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# Derivatives Cheat Sheet

• Algebra
• Number Rules
• Expand Rules
• Fractions Rules
• Absolute Rules
• Exponent Rules
• Radical Rules
• Factor Rules
• Factorial Rules
• Log Rules
• Undefined
• Complex Number Rules
• Trigonometry
• Basic Identities
• Pythagorean Identities
• Double-Angle Identities
• Sum/Difference Identities
• Product-To-Sum Identities
• Triple-Angle Identities
• Function Ranges
• Function Values
• Limits
• Limit Properties
• Limit to Infinity Properties
• Indeterminate Forms
• Common Limits
• Limit Rules
• Derivatives
• Derivatives Rules
• Common Derivatives
• Trigonometric Derivatives
• Arc Trigonometric Derivatives
• Hyperbolic Derivatives
• Arc Hyperbolic Derivatives
• Integrals
• Common Integrals
• Trigonometric Integrals
• Arc Trigonometric Integrals
• Hyperbolic Integrals
• Integrals of Special Functions
• Indefinite Integrals Rules
• Definite Integrals Rules

# Derivatives Cheat Sheet

## Derivatives Rules

Power Rule \frac{d}{dx}\left(x^a\right)=a\cdot x^{a-1}
Derivative of a constant \frac{d}{dx}\left(a\right)=0
Sum Difference Rule \left(f\pm g\right)^'=f^'\pm g^'
Constant Out \left(a\cdot f\right)^'=a\cdot f^'
Product Rule (f\cdot g)^'=f^'\cdot g+f\cdot g^'
Quotient Rule \left(\frac{f}{g}\right)^'=\frac{f^'\cdot g-g^'\cdot f}{g^2}
Chain rule \frac{df\left(u\right)}{dx}=\frac{df}{du}\cdot \frac{du}{dx}

## Common Derivatives

\frac{d}{dx}\left(\ln(x))=\frac{1}{x} \frac{d}{dx}\left(\ln(\left|x\right|))=\frac{1}{x}
\frac{d}{dx}\left(e^{x})=e^{x} \frac{d}{dx}\left(\log(x))=\frac{1}{x\ln(10)}
\frac{d}{dx}\left(\log_{a}(x))=\frac{1}{x\ln(a)}

## Trigonometric Derivatives

\frac{d}{dx}\left(\sin(x))=\cos(x) \frac{d}{dx}\left(\cos(x))=-\sin(x)
\frac{d}{dx}\left(\tan(x))=\sec^{2}(x) \frac{d}{dx}\left(\sec(x))=\frac{\tan(x)}{\cos(x)}
\frac{d}{dx}\left(\csc(x))=\frac{-\cot(x)}{\sin(x)} \frac{d}{dx}\left(\cot(x))=-\frac{1}{\sin^{2}(x)}

## Arc Trigonometric Derivatives

\frac{d}{dx}\left(\arcsin(x))=\frac{1}{\sqrt{1-x^{2}}} \frac{d}{dx}\left(\arccos(x))=-\frac{1}{\sqrt{1-x^{2}}}
\frac{d}{dx}\left(\arctan(x))=\frac{1}{x^{2}+1} \frac{d}{dx}\left(\arcsec(x))=\frac{1}{\sqrt{x^2}\sqrt{x^2-1}}
\frac{d}{dx}\left(\arccsc(x))=-\frac{1}{\sqrt{x^2}\sqrt{x^2-1}} \frac{d}{dx}\left(\arccot(x))=-\frac{1}{x^{2}+1}

## Hyperbolic Derivatives

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

## Arc Hyperbolic Derivatives

\frac{d}{dx}\left(\arcsinh(x))=\frac{1}{\sqrt{x^{2}+1}} \frac{d}{dx}\left(\arccosh(x))=\frac{1}{\sqrt{x-1}\sqrt{x+1}}
\frac{d}{dx}\left(\arctanh(x))=\frac{1}{1-x^2} \frac{d}{dx}\left(\arcsech(x))=\frac{\sqrt{\frac{2}{x+1}-1}}{(x-1)x}
\frac{d}{dx}\left(\arccsch(x))=-\frac{1}{\sqrt{\frac{1}{x^2}+1}x^2} \frac{d}{dx}\left(\arccoth(x))=\frac{1}{1-x^{2}}

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