Differentiation and Integration

Differentiation Rules

Constant. $\quad \dfrac{d}{dx} c = 0 .$

Constant Multiple. $\quad \dfrac{d}{dx} \big( c f(x) \big) = c f^\prime (x) .$

Sum. $\quad \dfrac{d}{dx} \big( f(x) \pm g(x) \big) = f^\prime (x) \pm g^\prime (x)$ .

Product. $\quad \dfrac{d}{dx} \big( f(x) g(x) \big) = f^\prime (x) g(x) + f(x) g^\prime (x)$ .

Quotient. $\quad \dfrac{d}{dx} \big( \dfrac{f(x)}{g(x)} \big) = \dfrac{f^\prime (x) g(x) - f(x) g^\prime (x) }{g^2(x)}$ .

Chain. $\quad \dfrac{d}{dx} f \big( g (x) \big) = f^\prime \big( g(x) \big) g^\prime (x)$ .

Power. $\quad \dfrac{d}{dx} x^n = n x^{n-1}$ .

Power. $\quad \dfrac{d}{dx} \big( g(x) \big)^n = n \big( g(x) \big)^{n-1} g^\prime (x)$ .

Trigonometric.

$\frac{d}{dx} \sin x = \cos x .$

$\frac{d}{dx} \cos x = - \sin x .$

$\frac{d}{dx} \tan x = \sec^2 x .$

$\frac{d}{dx} \cot x = - \csc^2 x .$

Inverse Trigonometric.

$\frac{d}{dx} \sin^{-1} x = \dfrac{1}{\sqrt{1-x^2}}.$

$\frac{d}{dx} \cos^{-1} x = - \dfrac{1}{\sqrt{1-x^2}} .$

$\frac{d}{dx} \tan^{-1} x = \dfrac{1}{1+x^2} .$

$\frac{d}{dx} \cot^{-1} x = - \dfrac{1}{1+x^2} .$

Hyperbolic.

$\frac{d}{dx} \sinh x = \cosh x .$

$\frac{d}{dx} \cosh x = \sinh x .$

$\frac{d}{dx} \tanh x = \dfrac{1}{\cosh ^2 x} .$

$\frac{d}{dx} \coth x = -\dfrac{1}{\sinh ^2 x}.$

Inverse Hyperbolic.

$\frac{d}{dx} \sinh^{-1} x = \dfrac{1}{\sqrt{x^2+1}} .$

$\frac{d}{dx} \cosh^{-1} x = \dfrac{1}{\sqrt{x^2-1}} .$

$\frac{d}{dx} \tanh^{-1} x = \dfrac{1}{1-x^2}, \quad |x| < 1 .$

$\frac{d}{dx} \coth^{-1} x = \dfrac{1}{1-x^2}, \quad |x| > 1 .$

Exponential.

$\frac{d}{dx} e^x = e^x .$

$\frac{d}{dx} b^x = b^x \big( \ln b \big) .$

Logarithmic.

$\frac{d}{dx} \ln |x| = \dfrac{1}{x} .$

$\frac{d}{dx} \log_b x = \dfrac{1}{x \big( \ln x \big)} .$

Of an Integral.

$\frac{d}{dx} \int_a^x g(t) dt = g(x) .$

$\frac{d}{dx} \int_a^b g(x,t) dt = \int_a^b \frac{\partial}{\partial x} g(x,t) dt .$

Integration Formulas

$\int u^n du = \dfrac{u^{n+1}}{n+1} + C$ .

$\int \dfrac{1}{u} du = \ln |u| + C .$

$\int e^u du = e^u + C.$

$\int b^u du = \dfrac{1}{\ln b} b^u + C .$

$\int \sin u du = - \cos u + C .$

$\int \cos u du = \sin u + C .$

$\int \sec^2 u du = \tan u + C .$

$\int \csc^2 u du = - \cot u + C .$

$\int \tan u du = - \ln | \cos u | + C .$

$\int \cot u du = \ln | \sin u | + C .$

$\int \sec u du = \ln \big( | \sec u + \tan u | \big) + C .$

$\int \csc u du = \ln \big( | \csc u - \cot u | \big) + C .$

$\int \sin^2 u du = \frac{1}{2} u - \frac{1}{4} \sin 2u + C .$

$\int \cos^2 u du = \frac{1}{2} u + \frac{1}{4} \sin 2u + C .$

$\int \dfrac{1}{\sqrt{a^2 - u^2}} du = \sin^{-1} \dfrac{u}{a} + C .$

$\int \dfrac{1}{\sqrt{a^2 + u^2}} du = \ln \big( | u + \sqrt{a^2 + u^2} | \big) + C .$

$\int \sqrt{a^2 - u^2} du = \dfrac{u}{2} \sqrt{a^2 - u^2} + \dfrac{a^2}{2} \sin^{-1} \dfrac{u}{a} + C .$

$\int \sqrt{a^2 + u^2} du = \dfrac{u}{2} \sqrt{a^2 + u^2} + \dfrac{a^2}{2} \ln \big( | u + \sqrt{a^2 + u^2} | \big) + C .$

$\int \dfrac{1}{a^2 - u^2} du = \dfrac{1}{a} \ln | \dfrac{a+u}{a-u} | + C .$

$\int \dfrac{1}{a^2 + u^2} du = \dfrac{1}{a} \tan^{-1} \dfrac{u}{a} + C .$

$\int \dfrac{1}{\sqrt{u^2 - a^2}} du = \ln | u+\sqrt{u^2-a^2} | + C .$

$\int \sqrt{u^2 - a^2} du = \dfrac{u}{2} \sqrt{u^2 - a^2} - \dfrac{a^2}{2} \ln | u + \sqrt{u^2 - a^2} | + C .$

Reference

Dennis G. Zill. Advanced Engineering Mathematics, $6^{th}$ edition. Jones $&$ Bartlett Learning. 2016.