1. Home
  2. Math Notes
  3. Calculus I
  4. Derivative

Table of Derivatives

Related calculator: Derivative Calculator

Below is the list of the most common derivatives.

f(x)f{{\left({x}\right)}} f′(x)f{'}\left({x}\right)
Power Rule
xnx^n nxn−1nx^{n-1}
Exponential Function
axa^x ln⁡(a)ax{\ln{{\left({a}\right)}}}{{a}}^{{x}}
ex{{e}}^{{x}} ex{{e}}^{{x}}
Logarithmic Function
log⁡a(x){\log}_{{a}}{\left({x}\right)} 1xln⁡(a)\frac{{1}}{{{x}{\ln{{\left({a}\right)}}}}}
ln⁡∣x∣{\ln}{\left|{x}\right|} 1x\frac{{1}}{{x}}
Trigonometric Functions
sin⁡(x){\sin{{\left({x}\right)}}} cos⁡(x){\cos{{\left({x}\right)}}}
cos⁡(x){\cos{{\left({x}\right)}}} −sin⁡(x)-{\sin{{\left({x}\right)}}}
tan⁡(x){\tan{{\left({x}\right)}}} 1cos⁡2(x)=sec⁡2(x)\frac{{1}}{{{{\cos}}^{{2}}{\left({x}\right)}}}={{\sec}}^{{2}}{\left({x}\right)}
cot⁡(x){\cot{{\left({x}\right)}}} −1sin⁡2(x)=−csc⁡2(x)-\frac{{1}}{{{{\sin}}^{{2}}{\left({x}\right)}}}=-{{\csc}}^{{2}}{\left({x}\right)}
sec⁡(x)=1cos⁡(x){\sec{{\left({x}\right)}}}=\frac{{1}}{{{\cos{{\left({x}\right)}}}}} sec⁡(x)tan⁡(x){\sec{{\left({x}\right)}}}{\tan{{\left({x}\right)}}}
csc⁡(x)=1sin⁡(x){\csc{{\left({x}\right)}}}=\frac{{1}}{{{\sin{{\left({x}\right)}}}}} −csc⁡(x)cot⁡(x)-{\csc{{\left({x}\right)}}}{\cot{{\left({x}\right)}}}
Inverse Trigonometric Functions
arcsin⁡(x){\operatorname{arcsin}{{\left({x}\right)}}} 11−x2\frac{{1}}{{\sqrt{{{1}-{{x}}^{{2}}}}}}
arccos⁡(x){\operatorname{arccos}{{\left({x}\right)}}} −11−x2-\frac{{1}}{{\sqrt{{{1}-{{x}}^{{2}}}}}}
arctan⁡(x){\operatorname{arctan}{{\left({x}\right)}}} 11+x2\frac{{1}}{{{1}+{{x}}^{{2}}}}
arccot(x)\text{arccot}{\left({x}\right)} −11+x2-\frac{{1}}{{{1}+{{x}}^{{2}}}}
arcsec(x)\text{arcsec}{\left({x}\right)} 1xx2−1\frac{{1}}{{{x}\sqrt{{{{x}}^{{2}}-{1}}}}}
arccsc(x)\text{arccsc}{\left({x}\right)} −1xx2−1-\frac{{1}}{{{x}\sqrt{{{{x}}^{{2}}-{1}}}}}
Hyperbolic Functions
sinh⁡(x){\sinh{{\left({x}\right)}}} cosh⁡(x){\cosh{{\left({x}\right)}}}
cosh⁡(x){\cosh{{\left({x}\right)}}} sinh⁡(x){\sinh{{\left({x}\right)}}}
tanh⁡(x){\tanh{{\left({x}\right)}}} 1cosh⁡2(x)=sech2(x)\frac{{1}}{{{{\cosh}}^{{2}}{\left({x}\right)}}}={\text{sech}}^{{2}}{\left({x}\right)}
coth⁡(x){\coth{{\left({x}\right)}}} −1sinh⁡2(x)=−csch⁡2(x)-\frac{{1}}{{{{\sinh}}^{{2}}{\left({x}\right)}}}=-{\operatorname{csch}}^{{2}}{\left({x}\right)}
sech(x)=1cosh⁡(x)\text{sech}{\left({x}\right)}=\frac{{1}}{{{\cosh{{\left({x}\right)}}}}} −sech(x)tanh⁡(x)-\text{sech}{\left({x}\right)}{\tanh{{\left({x}\right)}}}
csch⁡(x)=1sinh⁡(x)\operatorname{csch}{\left({x}\right)}=\frac{{1}}{{{\sinh{{\left({x}\right)}}}}} −csch⁡(x)coth⁡(x)-\operatorname{csch}{\left({x}\right)}{\coth{{\left({x}\right)}}}
Inverse Hyperbolic Functions
arcsinh(x)\text{arcsinh}{\left({x}\right)} 1x2+1\frac{{1}}{{\sqrt{{{{x}}^{{2}}+{1}}}}}
arccosh(x)\text{arccosh}{\left({x}\right)} 1x2−1\frac{{1}}{{\sqrt{{{{x}}^{{2}}-{1}}}}}
arctanh(x)\text{arctanh}{\left({x}\right)} 11−x2\frac{{1}}{{{1}-{{x}}^{{2}}}}
arccoth(x)\text{arccot}\text{h}{\left({x}\right)} 11−x2\frac{{1}}{{{1}-{{x}}^{{2}}}}
arcsech(x)\text{arcsec}\text{h}{\left({x}\right)} −1x1−x2-\frac{{1}}{{{x}\sqrt{{{1}-{{x}}^{{2}}}}}}
arccsch(x)\text{arccsc}\text{h}{\left({x}\right)} −1∣x∣1+x2-\frac{{1}}{{{\left|{x}\right|}\sqrt{{{1}+{{x}}^{{2}}}}}}
Differentiation Rules
c{c} 0{0}
g(x)+h(x){g{{\left({x}\right)}}}+{h}{\left({x}\right)} g′(x)+h′(x){g{'}}{\left({x}\right)}+{h}'{\left({x}\right)}
g(x)−h(x){g{{\left({x}\right)}}}-{h}{\left({x}\right)} g′(x)−h′(x){g{'}}{\left({x}\right)}-{h}'{\left({x}\right)}
c⋅g(x){c}\cdot{g{{\left({x}\right)}}} c⋅g′(x){c}\cdot{g{'}}{\left({x}\right)}
g(x)h(x){g{{\left({x}\right)}}}{h}{\left({x}\right)} g′(x)h(x)+g(x)h′(x){g{'}}{\left({x}\right)}{h}{\left({x}\right)}+{g{{\left({x}\right)}}}{h}'{\left({x}\right)}
g(x)h(x)\frac{{{g{{\left({x}\right)}}}}}{{{h}{\left({x}\right)}}} g′(x)h(x)−g(x)h′(x)h2(x)\frac{{{g{'}}{\left({x}\right)}{h}{\left({x}\right)}-{g{{\left({x}\right)}}}{h}'{\left({x}\right)}}}{{{{h}}^{{2}}{\left({x}\right)}}}
g(h(x)){g{{\left({h}{\left({x}\right)}\right)}}} g′(h(x))⋅h′(x){g{'}}{\left({h}{\left({x}\right)}\right)}\cdot{h}'{\left({x}\right)}
f−1(x){{f}}^{{-{1}}}{\left({x}\right)} 1f′(f−1(x))\frac{{1}}{{{f{'}}{\left({{f}}^{{-{1}}}{\left({x}\right)}\right)}}}