Derivatives and Integrals Cheat Sheet | Essential Calculus Guide

This Derivatives and Integrals Cheat Sheet provides quick reference formulas for common functions and their derivatives or integrals. It includes key rules like the power rule, product rule, chain rule, and integrals of basic functions, helping you solve calculus problems more efficiently. Keep this handy for quick problem-solving and revision!

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1. Derivatives:

Basic Properties/Formulas/Rules:

$\dfrac{d}{dx} \bigg( cf(x) \bigg) = c f'(x)$, where $c$ is any constant.

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

$\dfrac{d}{dx} \bigg( x^n \bigg) = n x^{n-1}$, $n$ is any number.

$\dfrac{d}{dx} \left( c \right) = 0$, $c$ is any constant.

$\frac{d}{dx} \bigg( e^{g(x)} \bigg) = g'(x) e^{g(x)}$

$\dfrac{d}{dx} \left[ \ln(g(x)) \right] = \dfrac{g'(x)}{g(x)}$

Product Rule

$\bigg( f(x)g(x) \bigg)' = f'(x)g(x) + f(x)g'(x)$

Quotient Rule

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

Chain Rule

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

Common Derivatives:

Polynomials

$\dfrac{d}{dx}(c) = 0, \ \ \ \ \ \dfrac{d}{dx}(x) = 1, \ \ \ \ \ \ \dfrac{d}{dx}(cx) = c$

$\dfrac{d}{dx}(x^n) = nx^{n-1} \ \ \ \ \ \ \ \ \dfrac{d}{dx}(cx^n) = ncx^{n-1}$

Trigonometric Functions

$\dfrac{d}{dx}[\sin(x)] = \cos(x)$

$\dfrac{d}{dx}[\cos(x)] = - \sin(x)$

$\dfrac{d}{dx}[\tan(x)] = \sec^2(x)$

$\dfrac{d}{dx}[\csc(x)] = - \csc(x) \cot(x)$

$\dfrac{d}{dx}[\sec(x)] = \sec(x) \tan(x)$

$\dfrac{d}{dx}[\cot(x)] = - \csc^2(x)$

Inverse Trigonometric Functions

$\dfrac{d}{dx}[\sin^{-1}(x)] = \dfrac{1}{\sqrt{1 - x^2}}$

$\dfrac{d}{dx}[\cos^{-1}(x)] = \dfrac{-1}{\sqrt{1 - x^2}}$

$\dfrac{d}{dx}[\tan^{-1}(x)] = \dfrac{1}{1 + x^2}$

$\dfrac{d}{dx}[\csc^{-1}(x)] = \dfrac{-1}{|x|\sqrt{x^2 - 1}}$

$\dfrac{d}{dx}[\sec^{-1}(x)] = \dfrac{1}{|x|\sqrt{x^2 - 1}}$

$\dfrac{d}{dx}[\cot^{-1}(x)] = \dfrac{-1}{1 + x^2}$

Exponential & Logarithmic Functions

$\dfrac{d}{dx}[a^x] = a^x \ln(a)$

$\dfrac{d}{dx}[e^x] = e^x$

$\dfrac{d}{dx}[\ln(x)] = \dfrac{1}{x}, x > 0$

$\dfrac{d}{dx}[\ln |x|] = \dfrac{1}{x}, x \neq 0$

$\dfrac{d}{dx}[\log_a(x)] = \dfrac{1}{x \ln(a)}, x > 0$

Hyperbolic Functions

$\dfrac{d}{dx}[\sinh(x)] = \cosh(x)$

$\dfrac{d}{dx}[\cosh(x)] = \sinh(x)$

$\dfrac{d}{dx}[\tanh(x)] = \text{sech}^2(x)$

$\dfrac{d}{dx}[csc h(x)] = - csch(x) \coth(x)$

$\dfrac{d}{dx}[sech(x)] = - sech(x) \tanh(x)$

$\dfrac{d}{dx}[\coth(x)] = - csch^2(x)$

2. Integrals:

Basic Properties/Formulas/Rules:

$\int cf(x) dx = c \int f(x) dx$, $c$ is a constant.

$\int f(x) \pm g(x) dx = \int f(x) dx \pm \int g(x) dx$

$\int_a^b f(x) dx = f(x) \Big|_a^b = F(b) - F(a)$, where $f(x) = \int f(x) dx$

$\int_a^b cf(x) dx = c \int_a^b f(x) dx$, $c$ is a constant.

$\int_a^b f(x) \pm g(x) dx = \int_a^b f(x) dx \pm \int_a^b g(x) dx$

$\int_a^a f(x) dx = 0$

$\int_a^b f(x) dx = -\int_b^a f(x) dx$

$\int_a^b f(x) dx = \int_a^c f(x) dx + \int_c^b f(x) dx$

$\int_a^b c dx = c(b - a)$, $c$ is a constant.

If $f(x) \geq 0$ on $a \leq x \leq b$ then $\int_a^b f(x) dx \geq 0$

If $f(x) \geq g(x)$ on $a \leq x \leq b$ then $\int_a^b f(x) dx \geq \int_a^b g(x) dx$

Common Integrals:

Polynomials

$\int dx = x + c$

$\int k dx = kx + c$

$\int x^n dx = \dfrac{1}{n + 1}x^{n+1} + c, n \neq -1$

$\int \dfrac{1}{x} dx = \ln|x| + c$

$\int x^{-1} dx = \ln|x| + c$

$\int x^{-n} dx = \dfrac{1}{-n + 1}x^{-n+1} + c, n \neq 1$

$\int \dfrac{1}{ax + b} dx = \dfrac{1}{a} \ln|ax + b| + c$

$\int x^\frac{p}{q} dx = \dfrac{1}{\frac{p}{q} + 1} x^{\frac{p}{q} + 1} + c = \dfrac{q}{p + q} x^{\frac{p + q}{q}} + c$

Trigonometric Functions

$\int \cos(u) du = \sin(u) + c$

$\int \sin(u) du = - \cos(u) + c$

$\int \sec^2u \space du = \tan(u) + c$

$\int \sec(u)\tan(u) du = \sec(u) + c$

$\int \csc(u) \cot(u) du = - \csc(u) + c$

$\int \csc^2(u) du = - \cot(u) + c$

$\int \tan(u) du = -\ln| \space \cos(u)| + c = \ln| \space \sec(u)| + c$

$\int \cot(u) du = \ln| \space \sin(u)| + c = -\ln| \space \csc(u)| + c$

$\int \sec(u) du = \ln| \space \sec(u) + \tan(u)| + c$

$\int \sec^3(u) du = \dfrac{1}{2}\left( \sec(u) \space \tan(u) + \ln| \space \sec(u) + \tan(u)| \right) + c$

$\int \csc(u) du = \ln| \space \csc(u) - \cot(u)| + c$

$\int \csc^3(u) du = \dfrac{1}{2} \left( - \space \csc(u) \space \cot(u) + \ln| \space \csc(u) - \cot(u)| \right) + c$

Exponential & Logarithmic Functions

$\int e^u du = e^u + c$

$\int a^u du = \dfrac{a^u}{\ln(a)} + c$

$\int \ln(u) du = u \ln(u) - u + c$

$\int e^{au} \sin(bu) du = \dfrac{e^{au}}{a^2 + b^2} \left( a \space \sin(bu) - b \space \cos(bu) \right) + c$

$\int ue^u du = (u - 1)e^u + c$

$\int e^{au} \cos(bu) du = \dfrac{e^{au}}{a^2 + b^2} \left( a\cos(bu) + b\sin(bu) \right) + c$

$\int \dfrac{1}{u\ln(u)} du = \ln|\ln(u)| + c$

Inverse Trigonometric Functions

$\int \dfrac{1}{\sqrt{a^2 - u^2}} du = \sin^{-1}\left( \frac{u}{a} \right) + c$

$\int \sin^{-1}(u) du = u \space \sin^{-1}(u) + \sqrt{1 - u^2} + c$

$\int \dfrac{1}{a^2 + u^2} du = \dfrac{1}{a} \space \tan^{-1}\left( \dfrac{u}{a} \right) + c$

$\int \tan^{-1}(u) du = u \space \tan^{-1}(u) - \dfrac{1}{2} \ln\left( 1 + u^2 \right) + c$

$\int \dfrac{1}{u \sqrt{u^2 - a^2}} du = \dfrac{1}{a} \space sec^{-1}\left( \dfrac{u}{a} \right) + c$

$\int \cos^{-1}(u) du = u \space \cos^{-1}(u) - \sqrt{1 - u^2} + c$

Hyperbolic Functions

$\int \sinh(u)du = \cosh(u) + c$

$\int sech(u) tanh(u)du = - sech(u) + c$

$\int sech^2(u)du = \tanh(u) + c$

$\int \cosh(u)du = \sinh(u) + c$

$\int csch(u) \coth(u)du = - csch(u) + c$

$\int csch^2(u)du = - \coth(u) + c$

$\int tanh(u)du = \ln(\cosh(u)) + c$

$\int sech(u)du = \tan^{-1}| \space \sinh(u)| + c$

Miscellaneous Integrals

$\int \dfrac{1}{a^2 - u^2} \ du = \dfrac{1}{2a} \ln\left| \dfrac{u + a}{u - a} \right| + c$

$\int \sqrt{a^2 + u^2} \ du = \dfrac{u}{2} \sqrt{a^2 + u^2} + \dfrac{a^2}{2} \ln\left| u + \sqrt{a^2 + u^2} \right| + c$

$\int \dfrac{1}{u^2 - a^2} \ du = \dfrac{1}{2a} \ln\left| \dfrac{u - a}{u + a} \right| + c$

$\int \sqrt{u^2 - a^2} \ du = \dfrac{u}{2} \sqrt{u^2 - a^2} - \dfrac{a^2}{2} \ln\left| u + \sqrt{u^2 - a^2} \right| + c$

$\int \sqrt{a^2 - u^2} \ du = \dfrac{u}{2} \sqrt{a^2 - u^2} + \dfrac{a^2}{2} \sin^{-1}\left( \dfrac{u}{a} \right) + c$

$\int \sqrt{2au - u^2} \ du = \dfrac{u - a}{2} \sqrt{2au - u^2} + \dfrac{a^2}{2} \cos^{-1}\left( \dfrac{a - u}{a} \right) + c$

Standard Integration Techniques:

$u$-Substitution

$\int_a^b f(g(x))g'(x) dx$ dx will convert the integral into $\int_a^b f(g(x))g'(x) dx = \int_{g(a)}^{g(b)} f(u) du$
using the substitution $u = g(x)$ where $du = g'(x)dx$. For indefinite integrals, drop the limits of integration.

Integration by Parts:

$\int u dv = uv - \int v du$ and $\int_a^b u dv = uv \Big|_a^b - \int_a^b v du$.
Choose $u$ and $dv$ from the integral and compute $du$ by differentiating $u$, and compute $v$ using $v = \int dv$.

Trigonometric Substitutions :

If the integral contains the following root use the given substitution and formula.

$\sqrt{a^2 - b^2x^2} \Rightarrow x = \dfrac{a}{b} \sin(\theta)$, and $\cos^2(\theta) = 1 - \sin^2(\theta)$

$\sqrt{b^2x^2 - a^2} \Rightarrow x = \dfrac{a}{b} \sec(\theta)$, and $\tan^2(\theta) = \sec^2(\theta) - 1$

$\sqrt{a^2 + b^2x^2} \Rightarrow x = \dfrac{a}{b} \tan(\theta)$, and $\sec^2(\theta) = 1 + \tan^2(\theta)$

Partial Fractions:

If integrating a rational expression involving polynomials, $\int \dfrac{P(x)}{Q(x)} dx$, where the degree (largest exponent) of $P(x)$ is smaller than the degree of $Q(x)$ then factor the denominator as completely as possible and find the partial fraction decomposition of the rational expression. Integrate the partial fraction decomposition (P.F.D.). We get terms in the decomposition for each factor in the denominator according to the following table.

| Factor of $Q(x)$ | Term in P.F.D. | Factor is $Q(x)$ | Term in P.F.D.

| --------------------- | --------------------| ----------------------l------------------------------------------

$\space \space \space ax + b \space \space \space \space \space \space \space \space \space \space \space \space \space \space\space \ \ \ \ \dfrac{A}{ax + b} \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \ \ \ \ (ax + b)^k \space \space \space \space \space \space \space\space \space \space\space \space \space \ \dfrac{A_1}{ax + b} + \dfrac{A_2}{(ax + b)^2} + \dots + \dfrac{A_k}{(ax + b)^k}$

$\space \space \space ax^2 + bx + c \space \space \space \space \dfrac{Ax + B}{ax^2 + bx + c} \space \space \space \space \space \space \space \space (ax^2 + bx + c)^k \space \space \space \space \space \dfrac{A_1x + B_1}{ax^2 + bx + c} + \dots + \dfrac{A_kx + B_k}{(ax^2 + bx + c)^k}$

Products and Quotients of Trigonometric Functions:

For $\int \sin^n(x) \space \cos^m(x) dx$ we have the following:

1. $n$ odd. strip 1 sine out and convert rest to cosines using $\sin^2(x) = 1 - \cos^2(x)$, Then use the substitution $u = \cos(x)$.
2. $m$ odd. Strip 1 cosine out and convert the rest to sines using $\cos^2(x) = 1 - \sin^2(x)$. Then, use the substitution $u = \sin(x)$.
3. $n$ and $m$ are both odd. use either 1. or 2.
4. $n$ and $m$ are both even. use double-angle and/or half-angle formulas to reduce the integral into an integrated form.

For $\int \tan^n(x) \sec^m(x) dx$ we have the following:

1. $n$ odd. strip 1 tangent and one secant, and convert the rest to secants using $\tan^2(x) = \sec^2(x) - 1$, Then use the substitution $u = \sec(x)$.
2. $m$ even. strip 2 secants out and convert rest to tangents using $\sec^2(x) = 1 + \tan^2(x)$, Then use the substitution $u = \tan(x)$.
3. $n$ odd and $m$ is even. use either 1. or 2.
4. $n$ is even and $m$ is odd. Each integral will be dealt with differently.

convert example:
$\sin^6(x)$ into $\left( \sin^2(x) \right)^3 = \left( 1 - \cos^2(x) \right)^3$

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