# Formulas and Rules for Derivative in Calculus



## Formulas

### $$\dfrac{d f(x)}{dx}$$

$$x^n$$ $$n x^{n-1}$$
$$e^x$$ $$e^x$$
$$b^x$$ $$\ln b \cdot b^x$$
$$\ln x$$ $$\dfrac{1}{x}$$
$$\log_b x$$ $$\dfrac{1}{ x \ln b}$$
$$\sin x$$ $$\cos x$$
$$\cos x$$ $$- \sin x$$
$$\tan x$$ $$\sec^2 x$$
$$\cot x$$ $$- \csc^2 x$$
$$\sec x$$ $$\sec x \tan x$$
$$\csc x$$ $$- \csc x \cot x$$
$$\sin^{-1} x$$ $$\dfrac{1}{\sqrt{1-x^2}}$$
$$\cos^{-1} x$$ $$- \dfrac{1}{\sqrt{1-x^2}}$$
$$\tan^{-1} x$$ $$\dfrac{1}{1+x^2}$$
$$\sinh x$$ $$\cosh x$$
$$\cosh x$$ $$\sinh x$$
$$\tanh x$$ $$\text{sech}^2 x$$
$$\coth x$$ $$- \text{csch}^2 x$$
$$\text{sech} \; x$$ $$-\text{sech} \; x \tanh x$$
$$\text{csch} \; x$$ $$- \text{csch} \; x \coth x$$
$$\sinh^{-1} x$$ $$\dfrac{1}{\sqrt{x^2+1}}$$
$$\cosh^{-1} x$$ $$\dfrac{1}{\sqrt{x^2-1}}$$
$$\tanh^{-1} x$$ $$\dfrac{1}{1-x^2}$$
$$\coth^{-1} x$$ $$\dfrac{1}{1-x^2}$$

## Rules

Let $$u$$ and $$v$$ be two functions with derivatives $$u'$$ and $$v'$$
• Linearity : $$\dfrac{d(a u + b v)}{dx} = a u' + b v'$$ , $$a$$ and $$b$$ are constants
• Derivative of Product : $$\dfrac{d(u v)}{dx} = u' v + u v'$$
• Derivative of Quotient : $$\dfrac{d\left(\dfrac{u}{v}\right)}{dx} = \dfrac{u' v - u v'}{v^2}$$
• Derivative of Composite : $$\dfrac{d(u (v) )}{dx} = \dfrac{du}{dv} \dfrac{dv}{dx}$$
• Derivative of a Function with a constant power: $$\dfrac{d(u^n)}{dx} = n u'u^{n-1}$$
• Derivative of a Function to the power of another function: $$\dfrac{d(u^v)}{dx} = \left(v' ln (u) + \dfrac{u'}{u} v \right) u^v$$