Calculus/Tables of Derivatives

General Rules

${\displaystyle {\frac {d}{dx}}(f+g)={\frac {df}{dx}}+{\frac {dg}{dx}}}$

${\displaystyle {\frac {d}{dx}}(cf)=c{\frac {df}{dx}}}$

${\displaystyle {\frac {d}{dx}}(fg)=f{\frac {dg}{dx}}+g{\frac {df}{dx}}}$

${\displaystyle {\frac {d}{dx}}\left({\frac {f}{g}}\right)={\frac {g{\frac {df}{dx}}-f{\frac {dg}{dx}}}{g^{2}}}}$

${\displaystyle [f(g(x))]'=f'(g(x))g'(x)}$

Powers and Polynomials

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

Trigonometric Functions

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

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

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

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

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

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

Exponential and Logarithmic Functions

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

Inverse Trigonometric Functions

• ${\displaystyle {\frac {d}{dx}}\arcsin x={\frac {1}{\sqrt {1-x^{2}}}}}$
• ${\displaystyle {\frac {d}{dx}}\arccos x=-{\frac {1}{\sqrt {1-x^{2}}}}}$
• ${\displaystyle {\frac {d}{dx}}\arctan x={\frac {1}{1+x^{2}}}}$
• ${\displaystyle {\frac {d}{dx}}\operatorname {arcsec} x={\frac {1}{|x|{\sqrt {x^{2}-1}}}}}$
• ${\displaystyle {\frac {d}{dx}}\operatorname {arccot} x={\frac {-1}{1+x^{2}}}}$
• ${\displaystyle {\frac {d}{dx}}\operatorname {arccsc} x={\frac {-1}{|x|{\sqrt {x^{2}-1}}}}}$

Hyperbolic and Inverse Hyperbolic Functions

${\displaystyle {d \over dx}\sinh x=\cosh x}$
${\displaystyle {d \over dx}\cosh x=\sinh x}$
${\displaystyle {d \over dx}\tanh x=\operatorname {sech} ^{2}\,x}$
${\displaystyle {d \over dx}\operatorname {sech} x=-\tanh x\,\operatorname {sech} x}$
${\displaystyle {d \over dx}\coth x=-\operatorname {csch} ^{2}x}$
${\displaystyle {d \over dx}\operatorname {csch} \,x=-\coth x\,\operatorname {csch} x}$
${\displaystyle {d \over dx}\sinh ^{-1}x={1 \over {\sqrt {x^{2}+1}}}}$
${\displaystyle {d \over dx}\cosh ^{-1}x={-1 \over {\sqrt {x^{2}-1}}}}$
${\displaystyle {d \over dx}\tanh ^{-1}x={1 \over 1-x^{2}}}$
${\displaystyle {d \over dx}\operatorname {sech} ^{-1}x={1 \over x{\sqrt {1-x^{2}}}}}$
${\displaystyle {d \over dx}\coth ^{-1}x={-1 \over 1-x^{2}}}$
${\displaystyle {d \over dx}\operatorname {csch} ^{-1}\,x={-1 \over |x|{\sqrt {1+x^{2}}}}}$