2.10 Table of Derivatives

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We learn the derivatives of many familiar functions.

Table of Derivatives

In this section we present the derivatives of the functions we have seen previously.

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

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

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

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