Derivative Rules

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See also Hyperbolic Identities.

Basic Functions

Let $a$ and $n$ be an arbitrary constants.

$$\begin{array}{rlrlr}& \frac{d}{dx}a& & =0& \\ & \frac{d}{dx}{x}^n& & =n{x}^{n-1}& \text{(power rule)}\end{array}$$

Exponential and Logarithmic Functions

Let $b$ be an arbitrary positive constant.

$$\begin{array}{rlrlr}& \frac{d}{dx}{e}^x& & =e^x& \\ & \frac{d}{dx}{b}^x& & =b^x\ln b& \\ & \frac{d}{dx}\ln x& & =\frac{1}{x}& \text{for }x>0\\ & \frac{d}{dx}{\log }_{b}x& & =\frac{1}{x\ln b}& \text{for }x>0\end{array}$$

Trigonometric Functions

$$\begin{array}{rlrl}& \frac{d}{dx}\sin x& & =\cos x\\ & \frac{d}{dx}\cos x& & =-\sin x\\ & \frac{d}{dx}\tan x& & ={\sec }^{2}x\\ & \frac{d}{dx}\cot x& & =-{\csc }^{2}x\\ & \frac{d}{dx}\sec x& & =\sec x\tan x\\ & \frac{d}{dx}\csc x& & =-\csc x\cot x\end{array}$$

Inverse Trigonometric Functions

Out of this list, the most important to remember are the formulas for the derivatives of $\arcsin$ and $\arctan$.

$$\begin{array}{rlrl}& \frac{d}{dx}\arcsin x& & =\frac{1}{\sqrt{1-x^2}}\\ & \frac{d}{dx}\arccos x& & =-\frac{1}{\sqrt{1-x^2}}\\ & \frac{d}{dx}\arctan x& & =\frac{1}{x^2+1}\\ & \frac{d}{dx}\text{arccot }x& & =-\frac{1}{x^2+1}\\ & \frac{d}{dx}\text{arcsec }x& & =\frac{1}{|x|\sqrt{x^2-1}}\\ & \frac{d}{dx}\text{arccsc }x& & =-\frac{1}{|x|\sqrt{x^2-1}}\end{array}$$

Hyperbolic Trigonometric Functions

Most calculus courses, including AP Calculus, do not cover these functions. You can ignore these if they are outside the scope of your class.

$$\begin{array}{rlrl}& \frac{d}{dx}\sinh x& & =\cosh x\\ & \frac{d}{dx}\cosh x& & =\sinh x\\ & \frac{d}{dx}\tanh x& & ={\text{sech}}^{2}x\\ & \frac{d}{dx}\text{coth }x& & =-{\text{csch}}^{2}x\\ & \frac{d}{dx}\text{sech }x& & =-\text{sech }x\tanh x\\ & \frac{d}{dx}\text{csch }x& & =-\text{csch }x\text{coth }x\end{array}$$

Inverse Hyperbolic Trigonometric Functions

Most calculus courses, including AP Calculus, do not cover these functions. You can ignore these if they are outside the scope of your class.

$$\begin{array}{rlrl}& \frac{d}{dx}\text{arcsinh }x& & =\frac{1}{\sqrt{x^2+1}}\\ & \frac{d}{dx}\text{arccosh }x& & =\frac{1}{\sqrt{x^2-1}}\\ & \frac{d}{dx}\text{arctanh }x& & =\frac{1}{1-x^2}\\ & \frac{d}{dx}\text{arccoth }x& & =\frac{1}{1-x^2}\\ & \frac{d}{dx}\text{arcsech }x& & =-\frac{1}{x\sqrt{1-x^2}}\\ & \frac{d}{dx}\text{arccsch }x& & =-\frac{1}{|x|\sqrt{1+x^2}}\end{array}$$

General Properties and Function Combinations

Let $f$ and $g$ be arbitrary functions and let $a$ and $b$ be arbitrary constants.

$$\begin{array}{rlrlr}& \frac{d}{dx}(af(x)\pm bg(x))& & =a{f}^{\prime }(x)\pm b{g}^{\prime }(x)& \text{(linearity)}\\ & \frac{d}{dx}f(x)g(x)& & =f^{\prime }(x)g(x)+f(x)g^{\prime }(x)& \text{(product rule)}\\ & \frac{d}{dx}\frac{f(x)}{g(x)}& & =\frac{f^{\prime }(x)g(x)-f(x)g^{\prime }(x)}{(g(x){)}^2}& \text{(quotient rule)}\\ & \frac{d}{dx}f(g(x))& & =f^{\prime }(g(x))g^{\prime }(x)& \text{(chain rule)}\\ & \frac{d}{dx}{f}^{-1}(x)& & =\frac{1}{f^{\prime }(f^{-1}(x))}& \text{(inverse func. thm.)}\end{array}$$