2.10 Table of Derivatives

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We learn the derivatives of many familiar functions.

1 Table of Derivatives

In this section, we present the derivatives of the functions introduced earlier.

$\begin {array}{c|c|c} Type & f(x) & f\,'(x) \\ \hline \text {Constant} & c & 0 \\[8pt] \text {Linear} & mx & m \\[8pt] \text {Power} & x^n & nx^{n-1} \\[8pt] \hline \text {Power} & \sqrt x & \displaystyle \frac {1}{2\sqrt x}\\[8pt] \text {Power} & \displaystyle \frac {1}{x} & -\displaystyle \frac {1}{x^2}\\[8pt] \hline \text {Trig} & \sin x & \cos x \\[8pt] \text {Trig} & \cos x & -\sin x \\[8pt] \hline \text {Trig} & \tan x & \sec ^2 x \\[8pt] \text {Trig} & \sec x & \sec x\tan x \\[8pt] \hline \text {Trig} & \cot x & -\csc ^2 x \\[8pt] \text {Trig} & \csc x & -\csc x\cot x \\[8pt] \hline \text {Inv Trig} & \sin ^{-1} x & \displaystyle {\frac {1}{\sqrt {1-x^2}}} \\[8pt] \text {Inv Trig} & \cos ^{-1} x & \displaystyle {-\frac {1}{\sqrt {1-x^2}}} \\[8pt] \text {Inv Trig} & \tan ^{-1} x & \displaystyle {\frac {1}{1+x^2}} \\[8pt] \hline \text {Exponential} & e^x & e^x \\[8pt] \text {Exponential} & a^x & a^x\ln a \\[8pt] \hline \text {Log} & \ln x & \displaystyle {\frac {1}{x}} \\[8pt] \text {Log} & \log _a x & \displaystyle {\frac {1}{x\ln a}} \end {array}$

The following table lists the chain rule versions of the above basic derivative formulas.

$\begin {array}{c|c|c} Type & f(x) & f\,'(x) \\ \hline \text {Power} & g^n(x) & ng^{n-1}(x)\cdot g\,'(x) \\[8pt] \hline \text {Trig} & \sin g(x) & \cos g(x) \cdot g\,'(x) \\[8pt] \text {Trig} & \cos g(x) & -\sin g(x)\cdot g\,'(x) \\[8pt] \hline \text {Power} & \sqrt {g(x)} & \displaystyle \frac {g'(x)}{2\sqrt {g(x)}}\\[8pt] \text {Power} & \displaystyle \frac {1}{g(x)} & -\displaystyle \frac {g\,'(x)}{g^2(x)}\\[8pt] \hline \text {Trig} & \tan g(x) & \sec ^2 g(x) \cdot g\,'(x) \\[8pt] \text {Trig} & \sec g(x) & \sec g(x) \tan g(x) \cdot g\,'(x) \\[8pt] \hline \text {Trig} & \cot g(x) & -\csc ^2 g(x) \cdot g\,'(x) \\[8pt] \text {Trig} & \csc g(x) & -\csc g(x) \cot g(x) \cdot g\,'(x) \\[8pt] \hline \text {Inv Trig} & \sin ^{-1} g(x) & \displaystyle \frac {g\,'(x)}{\sqrt {1-g^2(x)}} \\[8pt] \text {Inv Trig} & \cos ^{-1} g(x) & \displaystyle -\frac {g\,'(x)}{\sqrt {1-g^2(x)}}\\[8pt] \text {Inv Trig} & \tan ^{-1} g(x) & \displaystyle \frac {g\,'(x)}{1+g^2(x)} \\[8pt] \hline \text {Exponential} & e^{g(x)} & e^{g(x)} \cdot g\,'(x) \\[8pt] \text {Exponential} & a^{g(x)} & a^{g(x)}\ln a \cdot g\,'(x) \\[8pt] \hline \text {Log} & \ln g(x) & \displaystyle \frac {g\,'(x)}{g(x)} \\[8pt] \text {Log} & \log _a g(x) & \displaystyle \frac {g\,'(x)}{g(x)\ln a} \end {array}$

The following table is an alternative version of the above chain rule formulas with the variable $u$ representing a function of $x$ , i.e., $u = g(x)$ .

$\begin {array}{c|c|c} Type & f(x) & f\,'(x) \\ \hline \text {Power} & u^n & nu^{n-1}\cdot u\,' \\[8pt] \hline \text {Trig} & \sin (u) & \cos (u) \cdot u\,' \\[8pt] \text {Trig} & \cos (u) & -\sin (u)\cdot u\,' \\[8pt] \hline \text {Power} & \sqrt {u} & \displaystyle \frac {u\,' }{2\sqrt {u}}\\[8pt] \text {Power} & \displaystyle \frac {1}{u} & -\displaystyle \frac {u\,' }{u^2}\\[8pt] \hline \text {Trig} & \tan (u) & \sec ^2 (u) \cdot u\,' \\[8pt] \text {Trig} & \sec (u) & \sec (u)\tan (u) \cdot u\,' \\[8pt] \hline \text {Trig} & \cot (u) & -\csc ^2 (u) \cdot u\,' \\[8pt] \text {Trig} & \csc (u) & -\csc (u)\cot (u) \cdot u\,' \\[8pt] \hline \text {Inv Trig} & \sin ^{-1} (u) & \displaystyle \frac {u\,' }{\sqrt {1-u^2}} \\[8pt] \text {Inv Trig} & \cos ^{-1} (u) & \displaystyle -\frac {u\,' }{\sqrt {1-u^2}}\\[8pt] \text {Inv Trig} & \tan ^{-1} (u) & \displaystyle \frac {u\,' }{1+u^2} \\[8pt] \hline \text {Exponential} & e^{u} & e^{u} \cdot u\,' \\[8pt] \text {Exponential} & a^{u} & a^{u}\ln a \cdot u\,' \\[8pt] \hline \text {Log} & \ln (u) & \displaystyle \frac {u\,'}{u} \\[8pt] \text {Log} & \log _a (u) & \displaystyle \frac {u\,'}{u\ln a} \end {array}$

1.1 Differentiation Rules

Constant Multiple rule

$\frac {d}{dx} \left [ cf(x) \right ] = c f\,'(x)$

Sum/Difference Rule

$\frac {d}{dx} \left [ f(x) \pm g(x) \right ] = f\,'(x) \pm g\,'(x)$

Product Rule

$\frac {d}{dx} \left [ f(x) \cdot g(x) \right ] = f(x) \cdot g\,'(x) + f\,'(x) \cdot g(x)$

Quotient Rule

$\frac {d}{dx} \left [ \frac {f(x)}{g(x)} \right ] = \frac {g(x) \cdot f\,'(x) - f(x) \cdot g\,'(x)}{g^2(x)}$

Chain Rule

$\frac {d}{dx} \left [ f(g(x)) \right ] = f\,'(g(x)) \cdot g\,'(x)$

2026-03-21 03:08:33