0.2 Differentiation Rules

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We review the rules of differentiation.

1 Table of derivatives

In this section we present the derivatives of the fundamental functions in calculus.

$\begin {array}{c|c|c} Type & f(x) & f'(x) \\ \hline \text {Constant} & c & 0 \\ \text {Linear} & mx & m \\ \text {Power} & x^n & nx^{n-1} \\ \hline \text {Trig} & \sin x & \cos x \\ \text {Trig} & \cos x & -\sin x \\ \hline \text {Trig} & \tan x & \sec ^2 x \\ \text {Trig} & \sec x & \sec x\tan x \\ \hline \text {Trig} & \cot x & -\csc ^2 x\ \\ \text {Trig} & \csc x & -\csc x\cot x \\ \hline \text {Inv Trig} & \sin ^{-1} x & \frac {1}{\sqrt {1-x^2}} \\ \text {Inv Trig} & \cos ^{-1} x & -\frac {1}{\sqrt {1-x^2}} \\ \text {Inv Trig} & \tan ^{-1} x & \frac {1}{1+x^2} \\ \hline \text {Exponential} & e^x & e^x \\ \text {Exponential} & a^x & a^x\ln a \\ \hline \text {Log} & \ln x & \frac {1}{x} \\ \text {Log} & \log _a x & \frac {1}{x\ln a} \end {array}$

2 Combining functions

The following list shows how to find the derivative of a combination of functions.

1.: Constant multiple rule: $\frac {d}{dx} \left [cf(x)\right ] = c \frac {d}{dx} f(x),$ or, in condensed form: $(cf)' = cf'.$
2.: Sum and difference rules: $\frac {d}{dx} \left [f(x) \pm g(x)\right ] = \frac {d}{dx} f(x) \pm \frac {d}{dx} g(x),$ or, in condensed form: $(f \pm g)' = f' \pm g'.$
3.: Product rule: $\frac {d}{dx} \left [f(x)g(x)\right ] = f(x) \frac {d}{dx} g(x) + g(x) \frac {d}{dx} f(x),$ or, in condensed form: $(fg)' = fg' + gf'.$
4.: Quotient rule: $\frac {d}{dx} \left [\frac {f(x)}{g(x)}\right ] = \frac {g(x) \frac {d}{dx} f(x) - f(x) \frac {d}{dx} g(x)}{g^2(x)},$ or, in condensed form: $\left (\frac {f}{g}\right )' = \frac {gf' - fg'}{g^2}.$
5.: Chain rule: $\frac {d}{dx} \left [f(g(x))\right ] = f'(g(x)) \cdot g'(x),$ or, in Leibniz notation, if $y$ is a function of $u$ and $u$ is a function of $x$ , then $\frac {dy}{dx} = \frac {dy}{du} \cdot \frac {du}{dx}.$ In words, the derivative of a composition is the derivative of the outside times the derivative of the inside.

3 Problems

(problem 1) Find the derivative of $\displaystyle f(x) = \frac {1}{\sqrt x}$ $f'(x) = \answer {-1/2 x^{-3/2}}$

(problem 2) Find the derivative of $f(x) = x^2e^x$ $f'(x) = \answer {x^2e^x + 2xe^x}$

(problem 3) Find the derivative of $\displaystyle f(x) = \frac {x}{x^2+4}$ $f'(x) = \answer {\frac {4-x^2}{(x^2 + 4)^2}}$

(problem 4) Find the derivative of $f(x) = \cos (3x)$ $f'(x) = \answer {-3\sin (3x)}$

(problem 5) Find the derivative of $f(x) = e^{-2x}$ $f'(x) = \answer {-2e^{-2x}}$

(problem 6) Find the derivative of $\displaystyle f(x) =\frac {x^4}{8} + \frac {1}{8x^2}$ $f'(x) = \answer {\frac {x^3}{2} - \frac {1}{4x^3} }$

Examples of the differentiation rules can be found in Chapter 2 of the Calculus I textbook (also at ximera.osu.edu/math)

2024-09-27 13:57:45