Derivative Formula Sheet

Basic Derivative Formulas

Constant Rule: $\dfrac{d}{dx}C=0$

Constant Multiple Rule: $\dfrac{d}{dx}\Big(Cf(x)\Big)=C\dfrac{d}{dx}f(x)$

Sum and Difference Rule: $\dfrac{d}{dx}\Big(f(x)\pm g(x)\Big)=\dfrac{d}{dx}f(x)\pm\dfrac{d}{dx}g(x)$

Power Rule: $\dfrac{d}{dx}x^n=nx^{n-1}$

Product Rule:

$$\dfrac{d}{dx}\Big(f(x)g(x)\Big)=f(x)g'(x)+g(x)f'(x)$$

Quotient Rule:

$$\dfrac{d}{dx}\left(\dfrac{f(x)}{g(x)}\right)=\dfrac{g(x)f'(x)-f(x)g'(x)}{g(x)^2}$$

Chain Rule:

$$\dfrac{d}{dx}\Big(f(g(x))\Big)=f'(g(x))g'(x)$$

Where $f(x)$ is the outside function and $g(x)$ is the inside function.

Exponential and Logarithmic Derivatives

\[ \begin{array}{lll} \dfrac{d}{dx}e^x=e^x &&  \dfrac{d}{dx}a^x=a^x\ln{a} \\[10pt] \dfrac{d}{dx}\ln{x}=\dfrac{1}{x} && \dfrac{d}{dx}\log_a x=\dfrac{1}{x\ln{a}} \end{array} \]

Trigonometric Derivatives

\[ \begin{align*} &\dfrac d{dx} \sin x = \cos x \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \dfrac d{dx} \sec x = \sec x \tan x \\ &\dfrac d{dx} \cos x = -\sin x \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \dfrac d{dx} \csc x = -\csc x \tan x \\ &\dfrac d{dx} \tan x = \sec^2 x \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \dfrac d{dx} \cot x = -\csc x^2 \end{align*} \]

Inverse Trigonometric Derivatives

\[ \begin{align*} &\dfrac d{dx} \sin^{-1} x = \dfrac{1}{\sqrt{1-x^2}} \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \dfrac d{dx} \sec^{-1} x = \dfrac{1}{|x|\sqrt{x^2-1}} \\ &\dfrac d{dx} \cos^{-1} x = -\dfrac{1}{\sqrt{1-x^2}} \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \dfrac d{dx} \csc^{-1} x = -\dfrac{1}{|x|\sqrt{x^2-1}} \\ &\dfrac d{dx} \tan^{-1} x = \dfrac{1}{x^2+1} \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \dfrac d{dx} \cot^{-1} x = -\dfrac{1}{x^2+1} \end{align*} \]