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Math

Derivative Calculator

Differentiate any function symbolically. Supports nth-order derivatives and partial derivatives. Result in LaTeX with f(x) and f'(x) plot.

Table of derivatives

Derivatives of the elementary functions the calculator applies, for quick reference. In the steps above, each one shows up when it is used.

Function f(x)Derivative f'(x)
cc00
xnx^nn xn−1n\,x^{n-1}
exe^xexe^x
axa^xaxln⁡aa^x \ln a
ln⁡x\ln x1x\frac{1}{x}
log⁡ax\log_a x1xln⁡a\frac{1}{x \ln a}
sin⁡x\sin xcos⁡x\cos x
cos⁡x\cos x−sin⁡x-\sin x
tan⁡x\tan xsec⁡2x\sec^2 x
cot⁡x\cot x−csc⁡2x-\csc^2 x
sec⁡x\sec xsec⁡xtan⁡x\sec x \tan x
csc⁡x\csc x−csc⁡xcot⁡x-\csc x \cot x
arcsin⁡x\arcsin x11−x2\frac{1}{\sqrt{1-x^2}}
arccos⁡x\arccos x−11−x2-\frac{1}{\sqrt{1-x^2}}
arctan⁡x\arctan x11+x2\frac{1}{1+x^2}
x\sqrt{x}12x\frac{1}{2\sqrt{x}}
sinh⁡x\sinh xcosh⁡x\cosh x
cosh⁡x\cosh xsinh⁡x\sinh x

Differentiation rules

  • (f±g)′=f′±g′(f \pm g)' = f' \pm g'
  • (c f)′=c f′(c\,f)' = c\,f'
  • (f g)′=f′ g+f g′(f\,g)' = f'\,g + f\,g'
  • (fg)′=f′ g−f g′g2\left(\frac{f}{g}\right)' = \frac{f'\,g - f\,g'}{g^2}
  • (f(g(x)))′=f′(g(x))⋅g′(x)\left(f(g(x))\right)' = f'(g(x))\cdot g'(x)

Frequently Asked Questions

Clear your doubts and see practical examples

Use standard math syntax: sin(x)*x^2, e^(2*x), ln(x), 1/(1+x^2). Multiplication with *, power with ^.

Select "Order" and enter the desired number. We apply the derivative n times symbolically. Works for any positive integer n.

When f has more than one variable (e.g. x*y^2 + sin(x*y)), the partial derivative with respect to one variable treats the others as constants. Select "Partial" and choose the variable.

Yes. The symbolic engine applies all rules automatically: chain, product, quotient, derivatives of composite, trigonometric, exponential and logarithmic functions.

Yes. After calculating, we show f(x) (blue) and f'(x) (orange) overlaid so you can visualize critical points, monotonicity and concavity.