Derivative Formulas

General Rules

${\frac {\mathrm {d} }{\mathrm {d} x}}(f+g)={\frac {\mathrm {d} f}{\mathrm {d} x}}+{\frac {\mathrm {d} g}{\mathrm {d} x}}$

${\frac {\mathrm {d} }{\mathrm {d} x}}(c\cdot f)=c\cdot {\frac {\mathrm {d} f}{\mathrm {d} x}}$

${\frac {\mathrm {d} }{\mathrm {d} x}}(f\cdot g)=f\cdot {\frac {\mathrm {d} g}{\mathrm {d} x}}+g\cdot {\frac {\mathrm {d} f}{\mathrm {d} x}}$

${\frac {\mathrm {d} }{\mathrm {d} x}}\left({\frac {f}{g}}\right)={\dfrac {-f\cdot {\dfrac {\mathrm {d} g}{dx}}+g\cdot {\dfrac {\mathrm {d} f}{\mathrm {d} x}}}{g^{2}}}$

${\frac {\mathrm {d} }{\mathrm {d} x}}[f(g(x))]={\frac {\mathrm {d} f}{\mathrm {d} g}}\cdot {\frac {\mathrm {d} g}{\mathrm {d} x}}=f'(g(x))\cdot g'(x)$

${\frac {\mathrm {d} ^{n}}{\mathrm {d} x^{n}}}f(x)g(x)=\sum _{i=0}^{n}\left({\begin{matrix}n\\i\end{matrix}}\right)f^{(n-i)}(x)g^{(i)}(x)$

${\frac {\mathrm {d} }{\mathrm {d} x}}\left({\frac {1}{f}}\right)=-{\frac {f'}{f^{2}}}$

Powers and Polynomials

• ${\frac {\mathrm {d} }{\mathrm {d} x}}(c)=0$
• ${\frac {\mathrm {d} }{\mathrm {d} x}}x=1$
• ${\frac {\mathrm {d} }{\mathrm {d} x}}x^{n}=nx^{n-1}$
• ${\frac {\mathrm {d} }{\mathrm {d} x}}{\sqrt {x}}={\frac {1}{2{\sqrt {x}}}}$
• ${\frac {\mathrm {d} }{\mathrm {d} x}}{\frac {1}{x}}=-{\frac {1}{x^{2}}}$
• ${{\frac {\mathrm {d} }{\mathrm {d} x}}(c_{n}x^{n}+c_{n-1}x^{n-1}+c_{n-2}x^{n-2}+\cdots +c_{2}x^{2}+c_{1}x+c_{0})=nc_{n}x^{n-1}+(n-1)c_{n-1}x^{n-2}+(n-2)c_{n-2}x^{n-3}+\cdots +2c_{2}x+c_{1}}$

Trigonometric Functions

${\frac {\mathrm {d} }{\mathrm {d} x}}\sin(x)=\cos(x)$

${\frac {\mathrm {d} }{\mathrm {d} x}}\cos(x)=-\sin(x)$

${\frac {\mathrm {d} }{\mathrm {d} x}}\tan(x)=\sec ^{2}(x)$

${\frac {\mathrm {d} }{\mathrm {d} x}}\cot(x)=-\csc ^{2}(x)$

${\frac {\mathrm {d} }{\mathrm {d} x}}\sec(x)=\sec(x)\tan(x)$

${\frac {\mathrm {d} }{\mathrm {d} x}}\csc(x)=-\csc(x)\cot(x)$

Exponential and Logarithmic Functions

• ${\frac {\mathrm {d} }{\mathrm {d} x}}e^{x}=e^{x}$
• ${\frac {\mathrm {d} }{\mathrm {d} x}}a^{x}=a^{x}\ln(a)\qquad {\text{if }}a>0$
• ${\frac {\mathrm {d} }{\mathrm {d} x}}\ln(x)={\frac {1}{x}}$
• ${\frac {\mathrm {d} }{\mathrm {d} x}}\log _{a}(x)={\frac {1}{x\ln(a)}}\qquad {\text{if }}a>0\ ,\ a\neq 1$
• ${\frac {\mathrm {d} }{\mathrm {d} x}}(f^{g})={\frac {\mathrm {d} }{\mathrm {d} x}}\left(e^{g\ln(f)}\right)=f^{g}\left(f'{\frac {g}{f}}+g'\ln(f)\right)\ ,\qquad f>0$
• ${\frac {\mathrm {d} }{\mathrm {d} x}}(c^{f})={\frac {\mathrm {d} }{\mathrm {d} x}}\left(e^{f\ln(c)}\right)=c^{f}\ln(c)\cdot f'$

Inverse Trigonometric Functions

${\frac {\mathrm {d} }{\mathrm {d} x}}\arcsin(x)={\frac {1}{\sqrt {1-x^{2}}}}$

${\frac {\mathrm {d} }{\mathrm {d} x}}\arccos(x)=-{\frac {1}{\sqrt {1-x^{2}}}}$

${\frac {\mathrm {d} }{\mathrm {d} x}}\arctan(x)={\frac {1}{x^{2}+1}}$

${\frac {\mathrm {d} }{\mathrm {d} x}}\operatorname {arccot}(x)=-{\frac {1}{x^{2}+1}}$

${\frac {\mathrm {d} }{\mathrm {d} x}}\operatorname {arcsec}(x)={\frac {1}{|x|{\sqrt {x^{2}-1}}}}$

${\frac {\mathrm {d} }{\mathrm {d} x}}\operatorname {arccsc}(x)=-{\frac {1}{|x|{\sqrt {x^{2}-1}}}}$

Hyperbolic and Inverse Hyperbolic Functions

${\frac {\mathrm {d} }{\mathrm {d} x}}\sinh(x)=\cosh(x)$

${\frac {\mathrm {d} }{\mathrm {d} x}}\cosh(x)=\sinh(x)$

${\frac {\mathrm {d} }{\mathrm {d} x}}\tanh(x)={\rm {sech}}^{2}(x)$

${\frac {\mathrm {d} }{\mathrm {d} x}}{\rm {sech}}(x)=-\tanh(x){\rm {sech}}(x)$

${\frac {\mathrm {d} }{\mathrm {d} x}}\coth(x)=-{\rm {csch}}^{2}(x)$

${\frac {\mathrm {d} }{\mathrm {d} x}}{\rm {csch}}(x)=-\coth(x){\rm {csch}}(x)$

${\frac {\mathrm {d} }{\mathrm {d} x}}{\rm {arsinh}}(x)={\frac {1}{\sqrt {1+x^{2}}}}$

${\frac {\mathrm {d} }{\mathrm {d} x}}{\rm {arcosh}}(x)={\frac {1}{\sqrt {x^{2}-1}}}\ ,\ x>1$

${\frac {\mathrm {d} }{\mathrm {d} x}}{\rm {artanh}}(x)={\frac {1}{1-x^{2}}}\ ,\ |x|<1$

${\frac {\mathrm {d} }{\mathrm {d} x}}{\rm {arcsch}}(x)=-{\frac {1}{|x|{\sqrt {1+x^{2}}}}}\ ,\ x\neq 0$

${\frac {\mathrm {d} }{\mathrm {d} x}}{\rm {arsech}}(x)=-{\frac {1}{x{\sqrt {1-x^{2}}}}}\ ,\ 0<x<1$

${\frac {\mathrm {d} }{\mathrm {d} x}}{\rm {arcoth}}(x)={\frac {1}{1-x^{2}}}\ ,\ |x|>1$

Resources

• Table of Derivatives, Wikibooks: Calculus
• Derivative Function & Interpretations. PowerPoint file created by Professor Cynthia Roberts, UTSA.
• Limits & Derivative Formulas. PowerPoint file created by Professor Cynthia Roberts, UTSA.
• Exponential and Logarithms. PowerPoint file created by Professor Cynthia Roberts, UTSA.

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Content obtained and/or adapted from:

• Table of Derivatives, Wikibooks: Calculus under a CC BY-SA license