Unit 2 Beta

Differentiation: Definition and Fundamental Properties: the one-page sheet

2.1 Average and Instantaneous Rates of Change

An average rate of change is a change in output over a change in input, the slope of a secant line. Writing the second input as a + h gives the difference quotient. Letting h → 0 shrinks the interval to one instant, and the limit of the difference quotient, when it exists, is the instantaneous rate of change.

The average rate of change of $f$ on $[a, b]$ is $\frac{f(b) - f(a)}{b - a}$ : the change in output over the change in input, with units of output per input.
A secant line joins two points of a graph. Its slope is the average rate of change between them.
The difference quotient $\frac{f(a + h) - f(a)}{h}$ is the average rate on the interval from $a$ to $a + h$ ; $h$ can be negative.
The instantaneous rate of change at $x = a$ is $\lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$ , when that limit exists. Simplify the quotient first; substituting $h = 0$ at the start gives $\frac{0}{0}$ .

give a secant line whose slope is the average rate of change (f(b) − f(a))/(b − a) the average rate becomes the difference quotient (f(a + h) − f(a))/h the interval shrinks to the single instant x = a and the secant lines pivot toward one limiting line its value is the instantaneous rate of change of f at x = a

difference quotient: The average rate of change of <math class="calc-m"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> between <math class="calc-m"><semantics><mrow><mi>x</mi><mo>=</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">x = a</annotation></semantics></math> and <math class="calc-m"><semantics><mrow><mi>x</mi><mo>=</mo><mi>a</mi><mo>+</mo><mi>h</mi></mrow><annotation encoding="application/x-tex">x = a + h</annotation></semantics></math>: <math class="calc-m"><semantics><mrow><mfrac><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo>+</mo><mi>h</mi><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><mi>h</mi></mfrac></mrow><annotation encoding="application/x-tex">\frac{f(a + h) - f(a)}{h}</annotation></semantics></math> for <math class="calc-m"><semantics><mrow><mi>h</mi><mo>≠</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">h \ne 0</annotation></semantics></math>. Its limit as <math class="calc-m"><semantics><mrow><mi>h</mi><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">h \to 0</annotation></semantics></math> is the instantaneous rate of change at <math class="calc-m"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math>.
secant line: A line through two points of a graph. Its slope is the average rate of change of the function between those two inputs.

2.2 The Derivative and Its Notation

The derivative f′(a) is the limit of the difference quotient as h → 0 : the instantaneous rate of change of f at a and the slope of the tangent line there. Found at every x , it is the function f′(x) = dy/dx . The tangent line at a is y = f(a) + f′(a)(x − a) .

The derivative of $f$ at $a$ is $f'(a)$ $= \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$ , or equivalently $\lim_{x \to a} \frac{f(x) - f(a)}{x - a}$ . Both are the same limit of secant slopes.
The tangent line at $x = a$ passes through $(a, f(a))$ with slope $f'(a)$ : $y = f(a) + f'(a)(x - a)$ .
Done at every $x$ , the limit gives the derivative function $f'(x)$ . In Leibniz notation, $y = f(x)$ has derivative $\frac{dy}{dx}$ , and $\frac{d}{dx}$ means "the derivative of".
A derivative has the units of a rate: output units per input unit.

its value is called the derivative of f at a , written f′(a) they approach the tangent line at x = a , whose slope is f′(a) the result is a new function, f′(x) , also written y′ or dy/dx the tangent line is y = f(a) + f′(a)(x − a)

derivative: The instantaneous rate of change of a function. At <math class="calc-m"><semantics><mrow><mi>x</mi><mo>=</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">x = a</annotation></semantics></math>, <math class="calc-m"><semantics><mrow><mrow><mi>f</mi><mo lspace="0" rspace="0">′</mo></mrow><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f'(a)</annotation></semantics></math><math class="calc-m"><semantics><mrow><mo form="infix">=</mo><munder><mo movablelimits="false">lim</mo><mrow><mi>h</mi><mo>→</mo><mn>0</mn></mrow></munder><mfrac><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo>+</mo><mi>h</mi><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><mi>h</mi></mfrac></mrow><annotation encoding="application/x-tex">= \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}</annotation></semantics></math> when the limit exists; found at every <math class="calc-m"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>, it is the derivative function <math class="calc-m"><semantics><mrow><mrow><mi>f</mi><mo lspace="0" rspace="0">′</mo></mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f'(x)</annotation></semantics></math>.
tangent line: The line through <math class="calc-m"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a, f(a))</annotation></semantics></math> whose slope is <math class="calc-m"><semantics><mrow><mrow><mi>f</mi><mo lspace="0" rspace="0">′</mo></mrow><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f'(a)</annotation></semantics></math>: <math class="calc-m"><semantics><mrow><mi>y</mi><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>+</mo><mrow><mi>f</mi><mo lspace="0" rspace="0">′</mo></mrow><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y = f(a) + f'(a)(x - a)</annotation></semantics></math>. It is the limit of secant lines through that point.
Leibniz notation: Writing the derivative of <math class="calc-m"><semantics><mrow><mi>y</mi><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y = f(x)</annotation></semantics></math> as <math class="calc-m"><semantics><mrow><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{dy}{dx}</annotation></semantics></math>, and "the derivative of" as <math class="calc-m"><semantics><mrow><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{d}{dx}</annotation></semantics></math>. Its value at <math class="calc-m"><semantics><mrow><mi>x</mi><mo>=</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">x = a</annotation></semantics></math> is written <math class="calc-m"><semantics><mrow><msub><mrow><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo fence="true" stretchy="true">|</mo></mrow><mrow><mi>x</mi><mo>=</mo><mi>a</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\left.\frac{dy}{dx}\right|_{x=a}</annotation></semantics></math>.

2.3 Estimating a Derivative at a Point

With data instead of a formula, estimate f′(a) with a difference quotient from two table inputs on either side of a , as close as possible, and give the units. A symmetric quotient is the best simple estimate; a calculator's numerical derivative does the same with a tiny h , reported to three decimal places.

From a table, estimate $f'(a)$ by $\frac{f(b) - f(c)}{b - c}$ with $c < a < b$ : the two closest data points on either side of $a$ .
At the end of a table, only one side exists: use the closest one-sided quotient.
The symmetric difference quotient $\frac{f(a + h) - f(a - h)}{2h}$ averages the left and right quotients and is usually the most accurate.
A numerical derivative on a graphing calculator gives a decimal estimate of $f'(a)$ ; report it to three decimal places.
From a graph, read the slope of the tangent line from two points on it.

the limit in the definition cannot be taken, so use a difference quotient with the data their difference quotient is a good estimate of f′(a) the estimate is a rate with units, such as degrees per minute a calculator's numerical derivative estimates f′(a) with a tiny symmetric difference quotient

numerical derivative: An estimate of <math class="calc-m"><semantics><mrow><mrow><mi>f</mi><mo lspace="0" rspace="0">′</mo></mrow><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f'(a)</annotation></semantics></math> computed from values of <math class="calc-m"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> instead of a formula for <math class="calc-m"><semantics><mrow><mrow><mi>f</mi><mo lspace="0" rspace="0">′</mo></mrow></mrow><annotation encoding="application/x-tex">f'</annotation></semantics></math>: a difference quotient from a table, or a graphing calculator's built-in derivative, which uses <math class="calc-m"><semantics><mrow><mfrac><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo>+</mo><mi>h</mi><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo>−</mo><mi>h</mi><mo stretchy="false">)</mo></mrow><mrow><mn>2</mn><mi>h</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{f(a + h) - f(a - h)}{2h}</annotation></semantics></math> with a very small <math class="calc-m"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math>.

2.4 Differentiability and Continuity

A function is differentiable at a when f′(a) exists. Differentiability implies continuity, but not the other way round: corners, cusps and vertical tangents are continuous points with no derivative, and any discontinuity also rules a derivative out. At a break in a piecewise function, check continuity, then compare the one-sided slopes.

$f$ is differentiable at $a$ when $f'(a)$ exists: the limit of the difference quotient is a finite number.
Differentiable implies continuous. In a justification, say it: " $f$ is differentiable at $a$ , so $f$ is continuous at $a$ ."
Continuous does NOT imply differentiable. A continuous graph fails to have a derivative at a corner (one-sided slopes differ), a cusp (slopes become infinite in opposite directions) or a vertical tangent line.
Not continuous implies not differentiable: a jump, hole or asymptote rules out a derivative.
For a piecewise function at the break: check continuity first, then compare the left and right slopes.

f(a + h) − f(a) = (f(a + h) − f(a))/h ⋅h tends to f′(a)⋅0 = 0 lim as x → a, f(x) = f(a) : f is continuous at a the two-sided limit does not exist and f′(a) does not exist, even though f may be continuous there is no finite slope, so f′(a) does not exist

differentiable: A function is differentiable at <math class="calc-m"><semantics><mrow><mi>x</mi><mo>=</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">x = a</annotation></semantics></math> when <math class="calc-m"><semantics><mrow><mrow><mi>f</mi><mo lspace="0" rspace="0">′</mo></mrow><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f'(a)</annotation></semantics></math><math class="calc-m"><semantics><mrow><mo form="infix">=</mo><munder><mo movablelimits="false">lim</mo><mrow><mi>h</mi><mo>→</mo><mn>0</mn></mrow></munder><mfrac><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo>+</mo><mi>h</mi><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><mi>h</mi></mfrac></mrow><annotation encoding="application/x-tex">= \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}</annotation></semantics></math> exists as a finite number, and differentiable on an interval when it is differentiable at every point of it. Differentiable implies continuous; the converse is false.
corner: A point where a continuous graph has no derivative: a corner (the left and right slopes are different numbers, as for <math class="calc-m"><semantics><mrow><mo stretchy="false">|</mo><mi>x</mi><mo stretchy="false">|</mo></mrow><annotation encoding="application/x-tex">|x|</annotation></semantics></math> at 0), a cusp (the slopes grow without bound in opposite directions, as for <math class="calc-m"><semantics><mrow><msup><mi>x</mi><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow><annotation encoding="application/x-tex">x^{2/3}</annotation></semantics></math> at 0) or a vertical tangent line (as for <math class="calc-m"><semantics><mrow><mroot><mi>x</mi><mn>3</mn></mroot></mrow><annotation encoding="application/x-tex">\sqrt[3]{x}</annotation></semantics></math> at 0).

2.5 The Power Rule

The power rule says d/dx (x^r) = rx^(r − 1) for every real exponent r . Rewrite roots and fractions as powers of x first, subtract 1 carefully with negative and fractional exponents, and remember the rule needs the variable in the base, not in the exponent.

The power rule: $\frac{d}{dx}\left( x^r \right) = r x^{r - 1}$ for any real number $r$ . Bring the exponent down, then subtract 1 from it.
Rewrite first: $\sqrt{x} = x^{1/2}$ , $\frac{1}{x^4} = x^{-4}$ , $\frac{1}{\sqrt[3]{x}} = x^{-1/3}$ .
With negative exponents, "one less" means more negative: $-4 - 1 = -5$ .
Special cases: $\frac{d}{dx}(x) = 1$ , and the slope of $x^2$ at $x$ is $2x$ .
The rule needs the variable in the base and a fixed number in the exponent.

in each case the exponent comes down in front and the new exponent is one less the power rule: d/dx (x^r) = rx^(r − 1) rewrite it with a fractional or negative exponent before using the rule it is not a power of x , and the power rule does not apply

power rule: For any real number <math class="calc-m"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math>, <math class="calc-m"><semantics><mrow><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mrow><mo fence="true" stretchy="true">(</mo><msup><mi>x</mi><mi>r</mi></msup><mo fence="true" stretchy="true">)</mo></mrow><mo>=</mo><mi>r</mi><msup><mi>x</mi><mrow><mi>r</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\frac{d}{dx}\left( x^r \right) = r x^{r - 1}</annotation></semantics></math>: bring the exponent down and lower it by one. Roots and reciprocals are rewritten as powers first, such as <math class="calc-m"><semantics><mrow><msqrt><mi>x</mi></msqrt><mo>=</mo><msup><mi>x</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\sqrt{x} = x^{1/2}</annotation></semantics></math> and <math class="calc-m"><semantics><mrow><mfrac><mn>1</mn><msup><mi>x</mi><mn>2</mn></msup></mfrac><mo>=</mo><msup><mi>x</mi><mrow><mo form="prefix" lspace="0" rspace="0">−</mo><mn>2</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\frac{1}{x^2} = x^{-2}</annotation></semantics></math>.

2.6 Constant, Sum, Difference and Constant Multiple Rules

Constants added to a function disappear when you differentiate; constant factors stay. Derivatives of sums and differences are sums and differences of derivatives. With the power rule, these rules differentiate any polynomial term by term, and they work on tables of values as well as formulas.

Constant rule: $\frac{d}{dx}(c) = 0$ . Adding a constant shifts a graph up or down and changes no slope.
Constant multiple rule: $\frac{d}{dx}\left( kf(x) \right) = kf'(x)$ .
Sum and difference rules: $\frac{d}{dx}\left( f(x) \pm g(x) \right)$ $= f'(x) \pm g'(x)$ .
Together with the power rule: $\frac{d}{dx}\left( 4x^3 - 5x + 2 \right)$ $= 12x^2 - 5$ .
These rules work on values too: if $h = 3f - 2g$ , then $h'(2) = 3f'(2) - 2g'(2)$ .

its derivative is 0: d/dx (c) = 0 every slope is multiplied by k : (kf)′ = kf′ rates add: (f + g)′ = f′ + g′ , and (f − g)′ = f′ − g′ differentiate it term by term with the power rule

constant multiple rule: The basic derivative rules: the constant rule <math class="calc-m"><semantics><mrow><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\frac{d}{dx}(c) = 0</annotation></semantics></math>; the constant multiple rule <math class="calc-m"><semantics><mrow><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mrow><mo fence="true" stretchy="true">(</mo><mrow><mi>k</mi><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mo fence="true" stretchy="true">)</mo></mrow><mo>=</mo><mi>k</mi><mrow><mi>f</mi><mo lspace="0" rspace="0">′</mo></mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\frac{d}{dx}\left( kf(x) \right) = kf'(x)</annotation></semantics></math>; and the sum and difference rules <math class="calc-m"><semantics><mrow><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mrow><mo fence="true" stretchy="true">(</mo><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>±</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mo fence="true" stretchy="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">\frac{d}{dx}\left( f(x) \pm g(x) \right)</annotation></semantics></math><math class="calc-m"><semantics><mrow><mo form="infix">=</mo><mrow><mi>f</mi><mo lspace="0" rspace="0">′</mo></mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>±</mo><mrow><mi>g</mi><mo lspace="0" rspace="0">′</mo></mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">= f'(x) \pm g'(x)</annotation></semantics></math>.

2.7 Derivatives of sin x, cos x, e to the x and ln x

The derivatives of sine and cosine are cosine and negative sine (in radians); e^x is its own derivative; and ln x has derivative 1/x for x > 0 . With the sum and constant multiple rules they combine freely, and any limit of the form lim as h → 0, (f(a + h) − f(a))/h can be read as f′(a) and evaluated with them.

Learn the four: $\frac{d}{dx}(\sin x) = \cos x$ , $\frac{d}{dx}(\cos x) = -\sin x$ , $\frac{d}{dx}\left( e^x \right) = e^x$ , $\frac{d}{dx}(\ln x) = \frac{1}{x}$ for $x > 0$ .
The trigonometric formulas hold only in radians.
The power rule does not apply to $e^x$ : the variable is in the exponent.
A limit as a derivative: $\lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$ is $f'(a)$ . Name $f$ and $a$ , then use the derivative formula.

its slopes trace out the cosine curve: d/dx (sin x) = cos x its slopes are sine shifted the same way, which is −sin x the slope of e^x at every point equals its height: d/dx (e^x) = e^x a limit like lim as h → 0, (sin (a + h) − sin a)/h can be evaluated as cos a

derivative of sine: The four basic derivatives: <math class="calc-m"><semantics><mrow><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo stretchy="false">(</mo><mi>sin</mi><mo lspace="0" rspace="0.1667em">⁡</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>cos</mi><mo lspace="0" rspace="0.1667em">⁡</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">\frac{d}{dx}(\sin x) = \cos x</annotation></semantics></math> and <math class="calc-m"><semantics><mrow><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo stretchy="false">(</mo><mi>cos</mi><mo lspace="0" rspace="0.1667em">⁡</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo form="prefix" lspace="0" rspace="0">−</mo><mi>sin</mi><mo lspace="0" rspace="0.1667em">⁡</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">\frac{d}{dx}(\cos x) = -\sin x</annotation></semantics></math> (in radians), <math class="calc-m"><semantics><mrow><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mrow><mo fence="true" stretchy="true">(</mo><msup><mi>e</mi><mi>x</mi></msup><mo fence="true" stretchy="true">)</mo></mrow><mo>=</mo><msup><mi>e</mi><mi>x</mi></msup></mrow><annotation encoding="application/x-tex">\frac{d}{dx}\left( e^x \right) = e^x</annotation></semantics></math>, and <math class="calc-m"><semantics><mrow><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo stretchy="false">(</mo><mi>ln</mi><mo lspace="0" rspace="0.1667em">⁡</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow><annotation encoding="application/x-tex">\frac{d}{dx}(\ln x) = \frac{1}{x}</annotation></semantics></math> for <math class="calc-m"><semantics><mrow><mi>x</mi><mo>></mo><mn>0</mn></mrow><annotation encoding="application/x-tex">x > 0</annotation></semantics></math>.
limit as a derivative: Reading a limit of the form <math class="calc-m"><semantics><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi>h</mi><mo>→</mo><mn>0</mn></mrow></munder><mfrac><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo>+</mo><mi>h</mi><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><mi>h</mi></mfrac></mrow><annotation encoding="application/x-tex">\lim_{h \to 0} \frac{f(a + h) - f(a)}{h}</annotation></semantics></math> or <math class="calc-m"><semantics><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi>x</mi><mo>→</mo><mi>a</mi></mrow></munder><mfrac><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><mrow><mi>x</mi><mo>−</mo><mi>a</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\lim_{x \to a} \frac{f(x) - f(a)}{x - a}</annotation></semantics></math> as the derivative <math class="calc-m"><semantics><mrow><mrow><mi>f</mi><mo lspace="0" rspace="0">′</mo></mrow><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f'(a)</annotation></semantics></math>, and evaluating it with a derivative rule.

2.8 The Product Rule

When two factors both change, the product changes by each factor's change times the other factor: (fg)′ = f′g + fg′ . It is never just f′g′ . For three factors, differentiate one at a time and add the three terms. From a table, the rule needs the values as well as the derivatives.

The product rule: $\frac{d}{dx}\left( f(x)g(x) \right)$ $= f'(x)g(x)$ $+ f(x)g'(x)$ .
In words: derivative of the first times the second, plus the first times the derivative of the second.
The derivative of a product is NOT the product of the derivatives: $(x \cdot x)' = 2x$ , but $1 \cdot 1 = 1$ .
From a table: $(fg)'(a)$ $= f'(a)g(a) + f(a)g'(a)$ uses both the value columns and the derivative columns.
If one factor is a constant, the product rule reduces to the constant multiple rule.

the change in the product has two main parts: g times the change in f , and f times the change in g after dividing by Δx it still goes to 0 (fg)′ = f′g + fg′ differentiate one factor at a time and add: (fgk)′ = f′gk + fg′k + fgk′

product rule: <math class="calc-m"><semantics><mrow><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mrow><mo fence="true" stretchy="true">(</mo><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mo fence="true" stretchy="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">\frac{d}{dx}\left( f(x)g(x) \right)</annotation></semantics></math><math class="calc-m"><semantics><mrow><mo form="infix">=</mo><mrow><mi>f</mi><mo lspace="0" rspace="0">′</mo></mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">= f'(x)g(x)</annotation></semantics></math><math class="calc-m"><semantics><mrow><mo form="infix">+</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mrow><mi>g</mi><mo lspace="0" rspace="0">′</mo></mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">+ f(x)g'(x)</annotation></semantics></math>: the derivative of the first factor times the second, plus the first times the derivative of the second. It is not <math class="calc-m"><semantics><mrow><mrow><mi>f</mi><mo lspace="0" rspace="0">′</mo></mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mrow><mi>g</mi><mo lspace="0" rspace="0">′</mo></mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f'(x)g'(x)</annotation></semantics></math>.

2.9 The Quotient Rule

The derivative of f/g is (f′g − fg′)/g² , derived from the product rule by writing f = f/g ⋅g . Keep the numerator in order and square the denominator. When the denominator is a single term, dividing first and using the power rule is often simpler.

The quotient rule: $\frac{d}{dx}\left( \frac{f(x)}{g(x)} \right)$ $= \frac{f'(x)g(x) - f(x)g'(x)}{\left( g(x) \right)^2}$ , where $g(x) \ne 0$ .
Memory aid: "low d-high minus high d-low, over low squared."
The order in the numerator matters: swapping it flips the sign of the answer.
The derivative of a quotient is NOT the quotient of the derivatives.
For $\frac{3x^2 - 5}{x}$ , divide first: $3x - 5x^{-1}$ , then use the power rule.

the product rule gives f′ = q′g + qg′ q′ = (f′g − fg′)/g² the order matters: derivative of the top times the bottom comes first rewriting as a sum of powers is often faster than the quotient rule

quotient rule: <math class="calc-m"><semantics><mrow><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mrow><mo fence="true" stretchy="true">(</mo><mfrac><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac><mo fence="true" stretchy="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">\frac{d}{dx}\left( \frac{f(x)}{g(x)} \right)</annotation></semantics></math><math class="calc-m"><semantics><mrow><mo form="infix">=</mo><mfrac><mrow><mrow><mi>f</mi><mo lspace="0" rspace="0">′</mo></mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mrow><mi>g</mi><mo lspace="0" rspace="0">′</mo></mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><msup><mrow><mo fence="true" stretchy="true">(</mo><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mo fence="true" stretchy="true">)</mo></mrow><mn>2</mn></msup></mfrac></mrow><annotation encoding="application/x-tex">= \frac{f'(x)g(x) - f(x)g'(x)}{\left( g(x) \right)^2}</annotation></semantics></math> wherever <math class="calc-m"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≠</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">g(x) \ne 0</annotation></semantics></math>: derivative of the top times the bottom, minus the top times the derivative of the bottom, over the bottom squared.

2.10 Derivatives of tan, cot, sec and csc

Write tangent, cotangent, secant and cosecant in terms of sine and cosine, apply the quotient rule, and simplify with sin² x + cos² x = 1 . The results: (tan x)′ = sec² x , (cot x)′ = −csc² x , (sec x)′ = sec xtan x , (csc x)′ = −csc xcot x . The "co-" functions get the minus signs.

$\frac{d}{dx}(\tan x) = \sec^2 x$ and $\frac{d}{dx}(\cot x) = -\csc^2 x$ .
$\frac{d}{dx}(\sec x)$ $= \sec x\tan x$ and $\frac{d}{dx}(\csc x)$ $= -\csc x\cot x$ .
Every one comes from writing the function with $\sin x$ and $\cos x$ , using the quotient rule and the identity $\sin^2 x + \cos^2 x = 1$ .
The three "co-" functions (cosine, cotangent, cosecant) have derivatives with a minus sign.
All of these hold in radians, wherever the function is defined.

the quotient rule gives (cos xcos x − sin x(−sin x))/(cos² x) d/dx (tan x) = 1/(cos² x) = sec² x the quotient rule with a constant top gives sec xtan x and −csc xcot x its derivative carries a minus sign: −csc² x , −csc xcot x

derivative of tangent: The derivatives of the other four trigonometric functions: <math class="calc-m"><semantics><mrow><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo stretchy="false">(</mo><mi>tan</mi><mo lspace="0" rspace="0.1667em">⁡</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>sec</mi><mn>2</mn></msup><mo lspace="0" rspace="0.1667em">⁡</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">\frac{d}{dx}(\tan x) = \sec^2 x</annotation></semantics></math>, <math class="calc-m"><semantics><mrow><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo stretchy="false">(</mo><mi>cot</mi><mo lspace="0" rspace="0.1667em">⁡</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo form="prefix" lspace="0" rspace="0">−</mo><msup><mi>csc</mi><mn>2</mn></msup><mo lspace="0" rspace="0.1667em">⁡</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">\frac{d}{dx}(\cot x) = -\csc^2 x</annotation></semantics></math>, <math class="calc-m"><semantics><mrow><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo stretchy="false">(</mo><mi>sec</mi><mo lspace="0" rspace="0.1667em">⁡</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\frac{d}{dx}(\sec x)</annotation></semantics></math><math class="calc-m"><semantics><mrow><mo form="infix">=</mo><mi>sec</mi><mo lspace="0" rspace="0.1667em">⁡</mo><mi>x</mi><mi>tan</mi><mo lspace="0" rspace="0.1667em">⁡</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">= \sec x\tan x</annotation></semantics></math> and <math class="calc-m"><semantics><mrow><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo stretchy="false">(</mo><mi>csc</mi><mo lspace="0" rspace="0.1667em">⁡</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\frac{d}{dx}(\csc x)</annotation></semantics></math><math class="calc-m"><semantics><mrow><mo form="infix">=</mo><mo form="prefix" lspace="0" rspace="0">−</mo><mi>csc</mi><mo lspace="0" rspace="0.1667em">⁡</mo><mi>x</mi><mi>cot</mi><mo lspace="0" rspace="0.1667em">⁡</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">= -\csc x\cot x</annotation></semantics></math>, each found with the quotient rule from sine and cosine.