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Your calculus desk reference

Calculus formulas

Common derivatives, integrals, limits and multivariable formulas, with the conditions that make them valid. Open an example in the solver to explore the steps.

29 formulas found

Derivatives

Power

ddxxn=nxn−1\frac{d}{dx}x^n=nx^{n-1}

Use on intervals where the real power is defined and differentiable.

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Constant

ddxc=0\frac{d}{dx}c=0

c is constant with respect to x.

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Exponential

ddxex=ex\frac{d}{dx}e^x=e^x

Valid for all real x.

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General exponential

ddxax=axln⁡a\frac{d}{dx}a^x=a^x\ln a

a>0 is constant.

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Natural logarithm

ddxln⁡∣x∣=1x\frac{d}{dx}\ln|x|=\frac{1}{x}

x≠0; ln(x) itself requires x>0 over the reals.

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Sine

ddxsin⁡x=cos⁡x\frac{d}{dx}\sin x=\cos x

Angles are measured in radians.

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Cosine

ddxcos⁡x=−sin⁡x\frac{d}{dx}\cos x=-\sin x

Angles are measured in radians.

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Tangent

ddxtan⁡x=sec⁡2x\frac{d}{dx}\tan x=\sec^2 x

cos(x)≠0.

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Inverse sine

ddxarcsin⁡x=11−x2\frac{d}{dx}\arcsin x=\frac{1}{\sqrt{1-x^2}}

−1<x<1.

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Inverse tangent

ddxarctan⁡x=11+x2\frac{d}{dx}\arctan x=\frac{1}{1+x^2}

All real x.

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Integrals

Power integral

∫xn dx=xn+1n+1+C\int x^n\,dx=\frac{x^{n+1}}{n+1}+C

n≠−1, on an interval where xⁿ is defined.

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Reciprocal

∫1x dx=ln⁡∣x∣+C\int\frac{1}{x}\,dx=\ln|x|+C

x≠0; the integration interval must not cross zero.

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Exponential integral

∫ex dx=ex+C\int e^x\,dx=e^x+C

C is an arbitrary constant.

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Sine integral

∫sin⁡x dx=−cos⁡x+C\int\sin x\,dx=-\cos x+C

Angles in radians.

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Cosine integral

∫cos⁡x dx=sin⁡x+C\int\cos x\,dx=\sin x+C

Angles in radians.

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Inverse tangent integral

∫11+x2 dx=arctan⁡x+C\int\frac{1}{1+x^2}\,dx=\arctan x+C

All real x.

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Inverse sine integral

∫11−x2 dx=arcsin⁡x+C\int\frac{1}{\sqrt{1-x^2}}\,dx=\arcsin x+C

−1<x<1.

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Integration by parts

∫u dv=uv−∫v du\int u\,dv=uv-\int v\,du

Choose differentiable factors with the required integrals defined.

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Substitution

∫f(g(x))g′(x) dx=F(g(x))+C\int f(g(x))g'(x)\,dx=F(g(x))+C

F′=f and g is differentiable on the interval.

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Differentiation rules

Product rule

(uv)′=u′v+uv′(uv)'=u'v+uv'

u and v must be differentiable.

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Quotient rule

(uv)′=u′v−uv′v2\left(\frac{u}{v}\right)'=\frac{u'v-uv'}{v^2}

u and v differentiable; v≠0.

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Chain rule

ddxf(g(x))=f′(g(x))g′(x)\frac{d}{dx}f(g(x))=f'(g(x))g'(x)

Both component functions differentiable at the relevant points.

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Limits & theorems

Fundamental sine limit

lim⁡x→0sin⁡xx=1\lim_{x\to0}\frac{\sin x}{x}=1

Angles in radians.

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Exponential limit

lim⁡x→0ex−1x=1\lim_{x\to0}\frac{e^x-1}{x}=1

Two-sided real limit.

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L’Hôpital’s rule

lim⁡x→af(x)g(x)=lim⁡x→af′(x)g′(x)\lim_{x\to a}\frac{f(x)}{g(x)}=\lim_{x\to a}\frac{f'(x)}{g'(x)}

For 0/0 or ∞/∞ forms, with differentiability, nonzero g′ near a, and an existing derivative-ratio limit. Check the theorem’s hypotheses first.

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Fundamental theorem

ddx∫axf(t) dt=f(x)\frac{d}{dx}\int_a^x f(t)\,dt=f(x)

f continuous on an interval containing a and x.

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Multivariable calculus

Implicit derivative

F(x,y)=0⇒dydx=−FxFyF(x,y)=0\quad\Rightarrow\quad\frac{dy}{dx}=-\frac{F_x}{F_y}

F continuously differentiable near the point; F_y≠0.

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Partial power rule

∂∂x(xmyn)=mxm−1yn\frac{\partial}{\partial x}(x^m y^n)=m x^{m-1}y^n

Treat y as constant and stay in the differentiable domain.

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Equality of mixed partials

fxy=fyxf_{xy}=f_{yx}

A sufficient condition: both second partials are continuous in a neighborhood.

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