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Math Reference

Calculus References

Some useful formulas to help when doing calculus

Calculus

Special Functions

Exponential and Logarithmic Functions

log_a (x) = y \Leftrightarrow a^y = x
ln(x) = log_e (x)) where ln(e) = 1
ln (x) = y \Leftrightarrow e^y = x
Exponential and Logarithmic Functions
Cancellation Equations
log_a (a^x)=xa^{log_a (x)}=x
ln(e^x)=xe^{ln(x)}=x
Cancellation Equations
Laws of Logarithms
log_a (xy)=log_a (x) + log_a (y)
log_a (\frac{x}{y})=log_a (x) - log_a (y)
log_a (x^r)=rlog_a (x)
Laws of Logarithms

Hyperbolic Functions

sinh(x) = \frac{e^x - e^{-x}}{2}csch(x) = \frac{1}{sinh(x)}
cosh(x) = \frac{e^x + e^{-x}}{2}sech(x) = \frac{1}{cosh(x)}
tanh(x) = \frac{sinh(x)}{cosh(x)}coth(x) = \frac{cosh(x)}{sinh(x)}
Hyperbolic Functions

Inverse Hyperbolic Functions

y = sinh^{-1} (x) \Leftrightarrow sinh(y) = xsinh^{-1}(x) = ln(x + \sqrt{x^2 + 1})
y = cosh^{-1} (x) \Leftrightarrow cosh(y) = x) and y \geq 0cosh^{-1}(x) = ln(x + \sqrt{x^2 - 1})
y = tanh^{-1} (x) \Leftrightarrow tanh(y) = xtanh^{-1}(x) = \frac{1}{2} ln(\frac{1 + x}{1-x})
Inverse Hyperbolic Functions

Differentiation Rules

General Formulas

\frac{d}{dx} (c) = 0 where c is a constant\frac{d}{dx} (c f(x)) = cf^{\prime} (x) where c is a constant
\frac{d}{dx} (f(x) + g(x)) = f^{\prime} (x) + g^{\prime} (x)\frac{d}{dx} (f(x) - g(x)) = f^{\prime} (x) - g^{\prime} (x)
\frac{d}{dx} (f(x) g(x)) = f^{\prime} (x) g(x) + g^{\prime} (x) f(x)
(Product Rule)		\frac{d}{dx} (\frac{f(x)}{g(x)}) = \frac{f^{\prime} (x) g(x) - g^{\prime} (x) f(x)}{(g(x))^2}
(Quotient Rule)
\frac{d}{dx} f(g(x)) = f^{\prime} (g(x)) g^{\prime} (x)
(Chain Rule)		\frac{d}{dx} x^n = nx^{n-1}
(Power Rule)
General Formulas

Exponential and Logarithmic Functions

\frac{d}{dx} e^x = e^x\frac{d}{dx} a^x = a^x ln(a)
\frac{d}{dx} ln|x| = \frac{1}{x}\frac{d}{dx} log_a (x) = \frac{1}{x ln(a)}
Exponential and Logarithmic Functions

Trigonometric Functions

\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} csc(x) = -csc(x) cot(x)\frac{d}{dx} sec(x) = sec(x) tan(x)\frac{d}{dx} cot(x) = -csc^2 (x)
Trigonometric Functions

Inverse Trigonometric Functions

\frac{d}{dx} sin^{-1} (x) = \frac{1}{\sqrt{1 -x^2}}\frac{d}{dx} cos^{-1} (x) = - \frac{1}{\sqrt{1 -x^2}}\frac{d}{dx} tan^{-1} (x) = \frac{1}{1 + x^2}
\frac{d}{dx} csc^{-1} (x) = - \frac{1}{x \sqrt{x^2 - 1}}\frac{d}{dx} sec^{-1} (x) = \frac{1}{x \sqrt{x^2 - 1}}\frac{d}{dx} cot^{-1} (x) = - \frac{1}{1 + x^2}
Inverse Trigonometric Functions

Hyperbolic Functions

\frac{d}{dx} sinh(x) = cosh(x)\frac{d}{dx} cosh(x) = sinh(x)\frac{d}{dx} tanh(x) = sech^2 (x)
\frac{d}{dx} csch(x) = -csch(x) coth(x)\frac{d}{dx} sech(x) = -sech(x) tanh(x)\frac{d}{dx} coth(x) = -csch^2 (x)
Hyperbolic Functions

Inverse Hyperbolic Functions

\frac{d}{dx} sinh^{-1} (x) = \frac{1}{\sqrt{1 + x^2}}\frac{d}{dx} cosh^{-1} (x) = \frac{1}{\sqrt{x^2 - 1}}\frac{d}{dx} tanh^{-1} (x) = \frac{1}{1 - x^2}
\frac{d}{dx} csch^{-1} (x) = - \frac{1}{|x| \sqrt{x^2 + 1}}\frac{d}{dx} sech^{-1} (x) = - \frac{1}{x \sqrt{1 - x^2}}\frac{d}{dx} coth^{-1} (x) = \frac{1}{1 - x^2}
Inverse Hyperbolic Functions

Integrals

Let c be a constant

\int u dv = uv - \int v dv\int csc(u) cot(u) du = -csc(u) + c
\int u^n du = \frac{u^{n+1}}{n+1} + c where n\neq-1\int tan(u) du = ln|sec(u)| + c
\int \frac{du}{u} = ln|u| + c\int cot(u) du = ln|sin(u)| + c
\int e^u du = e^u +c\int sec(u) du = ln|sec(u) + tan(u)| + c
\int a^u du = \frac{a^u}{ln(a)} + c\int csc(u) du = ln|csc(u) - cot(u)| + c
\int sin(u) du = -cos(u) + c\int \frac{du}{\sqrt{a^2 - u^2}} du = sin^{-1} (\frac{u}{a}) + c
\int cos(u) du = sin(u) + c\int \frac{du}{a^2 + u^2} du = \frac{1}{a} tan^{-1} (\frac{u}{a}) + c
\int sec^2 (u) du = tan(u) + c\int \frac{du}{u \sqrt{u^2 - a^2}} du = \frac{1}{a} sec^{-1} (\frac{u}{a}) + c
\int csc^2 (u) du = -cot(u) + c\int \frac{du}{a^2 - u^2} du = \frac{1}{2a} ln|\frac{u+a}{u-a}| + c
\int sec(u) tan(u) du = sec(u) + c\int \frac{du}{u^2 - a^2} du = \frac{1}{2a} ln|\frac{u-a}{u+a}| + c
Integrals

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