Mathematics

Table of Derivatives

Comprehensive table of derivatives: differentiation rules, chain rule, elementary and composite functions.

y=Ky = Ky′=0y' = 0
y=f(x)+g(x)y = f(x) + g(x)y′=f′(x)+g′(x)y' = f'(x) + g'(x)
y=K⋅f(x)y = K \cdot f(x)y′=K⋅f′(x)y' = K \cdot f'(x)
y=f(x)⋅g(x)y = f(x) \cdot g(x)y′=f′(x)⋅g(x)+f(x)⋅g′(x)y' = f'(x) \cdot g(x) + f(x) \cdot g'(x)
y=f(x)g(x)\displaystyle y = \frac{f(x)}{g(x)}y′=f′(x)⋅g(x)−f(x)⋅g′(x)[g(x)]2\displaystyle y' = \frac{f'(x) \cdot g(x) - f(x) \cdot g'(x)}{[g(x)]^2}
y=f∘g(x)=f(g(x))y = f \circ g(x) = f(g(x))y′=dydx=dfdg⋅dgdx\displaystyle y' = \frac{dy}{dx} = \frac{df}{dg} \cdot \frac{dg}{dx}
y=xy = xy′=1y' = 1
y=xny = x^ny′=n⋅xn−1y' = n \cdot x^{n-1}
y=1x\displaystyle y = \frac{1}{x}y′=−1x2\displaystyle y' = -\frac{1}{x^2}
y=xy = \sqrt{x}y′=12x\displaystyle y' = \frac{1}{2\sqrt{x}}
y=xny = \sqrt[n]{x}y′=1n⋅xn−1n\displaystyle y' = \frac{1}{n \cdot \sqrt[n]{x^{n-1}}}
y=exy = e^xy′=exy' = e^x
y=axy = a^xy′=ax⋅ln⁡ay' = a^x \cdot \ln a
y=xxy = x^xy′=xx(1+ln⁡x)y' = x^x (1 + \ln x)
y=ln⁡xy = \ln xy′=1x\displaystyle y' = \frac{1}{x}
y=log⁡axy = \log_a xy′=1x⋅ln⁡a\displaystyle y' = \frac{1}{x \cdot \ln a}
y=sin⁡xy = \sin xy′=cos⁡xy' = \cos x
y=cos⁡xy = \cos xy′=−sin⁡xy' = -\sin x
y=tan⁡xy = \tan xy′=sec⁡2x=1cos⁡2x=1+tan⁡2x\displaystyle y' = \sec^2 x = \frac{1}{\cos^2 x} = 1 + \tan^2 x
y=arcsin⁡xy = \arcsin xy′=11−x2\displaystyle y' = \frac{1}{\sqrt{1-x^2}}
y=arccos⁡xy = \arccos xy′=−11−x2\displaystyle y' = \frac{-1}{\sqrt{1-x^2}}
y=arctan⁡xy = \arctan xy′=11+x2\displaystyle y' = \frac{1}{1+x^2}