Summary of Derivatives
Let $c$ be any constant. Let the derivative of a function $f(x)$ be denoted by $\frac{d}{dx} f(x) = f’(x)$. The following are general properties of the derivative operation
| Function | Derivative |
|---|---|
| $c \cdot f(x)$ | $c \cdot f’(x)$ |
| $f(x) \pm g(x)$ | $f’(x) \pm g’(x)$ |
| $f(x) \cdot g(x)$ | $f’(x) \cdot g(x) + f(x) \cdot g’(x)$ |
| $\frac{1}{f(x)}$ | $-\frac{f’(x)}{(f(x))^2}$ |
| $\frac{f(x)}{g(x)}$ | $\frac{f’(x) \cdot g(x) - f(x) \cdot g’(x)}{(g(x))^2}$ |
| $f^{-1}(x)$ | $\frac{1}{f’(f^{-1}(x))}$ |
| $f(g(x))$ | $f’(g(x)) \cdot g’(x)$ |
Let the $n$th derivative of a function $f(x)$ be denoted by $\frac{d^n}{dx^n} f(x) = f^{(n)}(x)$. An additional less commonly used, but interesting property
\[\frac{d^n}{dx^n} [f(x) \cdot g(x)] = \sum_{k=0}^{n} \binom{n}{k} f^{(n-k)}(x) g^{(k)}(x)\]Let $c$ and $r$ be any constant. Let $b > 1$ be any constant. Let $e$ be Euler’s number and $\ln x = \log_e x$ be the natural logarithm. The following are derivatives of some particular functions.
| Function | Derivative |
|---|---|
| $c$ | $0$ |
| $x$ | $1$ |
| $x^r$ | $r x^{r-1}$ |
| $\frac{1}{x}$ | $-\frac{1}{x^2}$ |
| $\sqrt{x}$ | $\frac{1}{2 \sqrt{x}}$ |
| $\lvert x \rvert$ | $\frac{x}{\lvert x \rvert}$ |
| $b^x$ | $b^x \ln b$ |
| $e^x$ | $e^x$ |
| $\log_b x$ | $\frac{1}{x \ln b}$ |
| $\ln x$ | $\frac{1}{x}$ |
We have derivatives of the trigonometric and inverse trigonometric functions.
| Function | Derivative | Condition |
|---|---|---|
| $\sin x$ | $\cos x$ | |
| $\cos x$ | $- \sin x$ | |
| $\tan x$ | $\sec^2 x$ | |
| $\sec x$ | $\tan x \sec x$ | |
| $\csc x$ | $- \cot x \csc x$ | |
| $\cot x$ | $-\csc^2 x$ | |
| $\arcsin x$ | $\frac{1}{\sqrt{1 - x^2}}$ | $\abs{x} \leq 1 $ |
| $\arccos x$ | $\frac{- 1}{\sqrt{1 - x^2}}$ | $\abs{x} \leq 1 $ |
| $\arctan x$ | $\frac{1}{1 + x^2}$ | |
| $\arcsec x$ | $\frac{1}{\lvert x \rvert \sqrt{x^2 - 1} }$ | $x \geq 1 $ |
| $\arccsc x$ | $\frac{- 1}{\lvert x \rvert \sqrt{x^2 - 1} }$ | $x \geq 1 $ |
| $\arccot x$ | $\frac{- 1}{1 + x^2}$ |
We have derivatives of the hyperbolic trigonometric and inverse hyperbolic trigonometric functions.
| Function | Derivative | Condition |
|---|---|---|
| $\sinh x$ | $\cosh x$ | |
| $\cosh x$ | $\sinh x$ | |
| $\tanh x$ | $\sech^2 x$ | |
| $\sech x$ | $-\tanh x \sech x$ | |
| $\csch x$ | $- \coth x \csch x$ | $x \neq 0$ |
| $\coth x$ | $- \csch^2 x$ | $x \neq 0$ |
| $\arcsinh x$ | $\frac{1}{\sqrt{1 + x^2}}$ | |
| $\arccosh x$ | $\frac{1}{\sqrt{x^2 - 1}}$ | $x > 1$ |
| $\arctanh x$ | $\frac{1}{1-x^2}$ | $\lvert x \rvert < 1$ |
| $\arcsech x$ | $\frac{- 1}{\lvert x \rvert \sqrt{1 + x^2} }$ | $0 < x < 1$ |
| $\arccsch x$ | $\frac{- 1}{\lvert x \rvert \sqrt{1 - x^2} }$ | $x \neq 0$ |
| $\arccoth x$ | $\frac{1}{1-x^2}$ | $\lvert x \rvert > 1$ |