## Table of Derivatives

## Handy Table of Derivatives

Want lots of examples to see *how * to calculate derivatives? Visit our free Calculating Derivatives: Problems & Solutions page!

### Power of *x*

\begin{align*}

\frac{d}{dx} \left(c\right) &= 0 \\[8px]
\frac{d}{dx} \left(cx\right) &= c \\[8px]
\frac{d}{dx} \left(cx^n\right) &= ncx^{n-1} \\[8px]
\end{align*}

For example,

\[\dfrac{d}{dx}5x^3 = 3 \cdot 5x^2 = 15x^2 \]

You’ll also want to remember that $\dfrac{1}{x^n} = x^{-n}$ (for example, $\dfrac{1}{x^2} = x^{-2}),$ and $\sqrt[n]{x} = x^{1/n}$ (example: $\sqrt[3]{x} = x^{1/3}$).

### Exponential and Logarithmic

\begin{align*}

\frac{d}{dx}\left( e^x \right) &= e^x &&& \frac{d}{dx}\left( a^x \right) &= a^x \ln a \\ \\

\frac{d}{dx}\left( \ln x \right) &= \frac{1}{x} &&& \frac{d}{dx}\left( \log_a x \right) &= \frac{1}{x \ln a}

\end{align*}

### Trigonometric

\begin{align*}

\frac{d}{dx}\left(\sin x\right) &= \cos x &&& \frac{d}{dx}\left(\csc x\right) &= -\csc x \cot x \\ \\

\dfrac{d}{dx}\left(\cos x\right) &= -\sin x &&& \frac{d}{dx}\left(\sec x\right) &= \sec x \tan x \\ \\

\dfrac{d}{dx}\left(\tan x\right) &= \sec^2 x &&& \frac{d}{dx}\left(\cot x\right) &= -\csc^2 x

\end{align*}

### Inverse Trigonometric

\begin{align*}

\frac{d}{dx}\left(\sin^{-1} x\right) &= \frac{1}{\sqrt{1 – x^2}} &&& \frac{d}{dx}\left(\csc^{-1} x\right) &= \frac{-1}{x\sqrt{x^2 – 1}} \\ \\

\dfrac{d}{dx}\left(\cos^{-1} x\right) &= \frac{-1}{\sqrt{1 – x^2}} &&& \frac{d}{dx}\left(\sec^{-1} x\right) &= \frac{1}{x\sqrt{x^2 – 1}} \\ \\

\dfrac{d}{dx}\left(\tan^{-1} x\right) &= \frac{1}{1 + x^2} &&& \frac{d}{dx}\left(\cot^{-1} x\right) &= \frac{-1}{1+x^2}

\end{align*}

### Hyperbolic

\begin{align*}

\text{Reminder:} \\

\sinh x &= \frac{e^x – e^{-x}}{2} & \cosh x &= \frac{e^x + e^{-x}}{2} & \tanh x &= \frac{\sinh x}{\cosh x} \\ \\

\text{csch }x &= \frac{1}{\sinh x} & \text{sech }x &= \frac{1}{\cosh x} & \coth x &= \frac{\cosh x}{\sinh x} \\

\end{align*}

\begin{align*}

\frac{d}{dx}\left(\sinh x\right) &= \cosh x &&& \frac{d}{dx}\left(\text{csch } x\right) &= -\text{csch } x \coth x \\ \\

\dfrac{d}{dx}\left(\cosh x\right) &= \sinh x &&& \frac{d}{dx}\left(\text{sech } x\right) &= -\text{sech } x \tanh x \\ \\

\dfrac{d}{dx}\left(\tanh x\right) &= \text{sech}^2 x &&& \frac{d}{dx}\left(\coth x\right) &= -\text{csch}^2 x

\end{align*}

*Tip:* You can differentiate any function, for free,

using Wolfram *WolframAlpha’s* Online Derivative Calculator.