Calculus/Tables of Derivatives

 ← Summation notation Calculus Tables of Integrals → Tables of Derivatives

General RulesEdit

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

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

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

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

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

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

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

Powers and PolynomialsEdit

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

Trigonometric FunctionsEdit

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

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

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

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

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

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

Exponential and Logarithmic FunctionsEdit

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

Inverse Trigonometric FunctionsEdit

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

Hyperbolic and Inverse Hyperbolic FunctionsEdit

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

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

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

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

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

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

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

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

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

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

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

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