Derivatives Cheat Sheet


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Derivatives Rules

Power Rule \frac{d}{dx}\left(x^a\right)=a\cdot x^{a-1} \:\frac{d}{dx}\left(x^3\right)
Derivative of a constant \frac{d}{dx}\left(a\right)=0 \:\frac{d}{dx}\left(5\right)
Sum Difference Rule \left(f\pm g\right)^'=f^'\pm g^' \:\frac{d}{dx}\left(x^3+\sqrt{x}\right)
Constant Out \left(a\cdot f\right)^'=a\cdot f^' \:\frac{d}{dx}\left(5x^3\right)
Product Rule (f\cdot g)^'=f^'\cdot g+f\cdot g^' \frac{d}{dx}(x(2x+3))
Quotient Rule \left(\frac{f}{g}\right)^'=\frac{f^'\cdot g-g^'\cdot f}{g^2} \frac{d}{dx}(\frac{x}{1+x})
Chain rule \frac{df\left(u\right)}{dx}=\frac{df}{du}\cdot \frac{du}{dx} \left(\sin ^2\left(θ\right)\right)^{'\:}


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(x^2-1)}}
\frac{d}{dx}\left(\arccsc(x))=-\frac{1}{\sqrt{x^2-1}\left|x\right|} \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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