Factsheet: List of derivatives
Mathematics
Summary
A list of common (and some uncommon) derivatives of functions.
Throughout, \(a,k\) are real numbers.
Derivatives of polynomial, exponential and logarithmic functions
| function | derivative w.r.t \(x\) | notes |
|---|---|---|
| \(c\) | \(0\) | \(c\in\mathbb{R}\) |
| \(mx + c\) | \(m\) | \(m,c\in\mathbb{R}\) |
| \(x^\alpha\) | \(\alpha x^{\alpha-1}\) | \(\alpha \in\mathbb{R}, \alpha \neq 0\) |
| \(ae^{kx}\) | \(ake^{kx}\) | |
| \(a\ln(kx)\) | \(\dfrac{a}{x}\) | |
| \(ac^{kx}\) | \(akc^{kx}\ln(b)\) | \(c \in\mathbb{R}, c > 0\) constant |
| \(a\log_c(kx)\) | \(\dfrac{a}{x\ln(c)}\) | \(c \in\mathbb{R}, c > 1\) constant |
Derivatives of trigonometric functions
| function | derivative w.r.t \(x\) |
|---|---|
| \(a\sin(kx)\) | \(ak\cos(kx)\) |
| \(a\cos(kx)\) | \(-ak\sin(kx)\) |
| \(a\tan(kx)\) | \(ak\sec^2(kx)\) |
| \(a\cot(kx)\) | \(-ak\csc^2(kx)\) |
| \(a\sec(kx)\) | \(ak\sec(kx)\tan(kx)\) |
| \(a\csc(kx)\) | \(-ak\csc(kx)\cot(kx)\) |
Derivatives of inverse trigonometric functions
| function | derivative w.r.t \(x\) | notes |
|---|---|---|
| \(a\sin^{-1}(kx)\) | \(\displaystyle\frac{ak}{\sqrt{1-k^2x^2}}\) | valid for \(x\in\left(-\frac{1}{k},\frac{1}{k}\right)\) |
| \(a\cos^{-1}(kx)\) | \(\displaystyle\frac{-ak}{\sqrt{1-k^2x^2}}\) | valid for \(x\in\left(-\frac{1}{k},\frac{1}{k}\right)\) |
| \(a\tan^{-1}(kx)\) | \(\displaystyle\frac{ak}{1+k^2x^2}\) | valid for \(x\in\mathbb{R}\) |
| \(a\cot^{-1}(kx)\) | \(\displaystyle\frac{-ak}{1+k^2x^2}\) | valid for \(x\in\mathbb{R}\) |
| \(a\sec^{-1}(kx)\) | \(\displaystyle\frac{a}{|x|\sqrt{k^2x^2 - 1}}\) | valid for \(x\in\mathbb{R}\smallsetminus\left(-\frac{1}{k},\frac{1}{k}\right)\) |
| \(a\csc^{-1}(kx)\) | \(\displaystyle\frac{-a}{|x|\sqrt{k^2x^2 - 1}}\) | valid for \(x\in\mathbb{R}\smallsetminus\left(-\frac{1}{k},\frac{1}{k}\right)\) |
Derivatives of hyperbolic functions
| function | derivative w.r.t \(x\) |
|---|---|
| \(a\sinh(kx)\) | \(ak\cosh(kx)\) |
| \(a\cosh(kx)\) | \(ak\sinh(kx)\) |
| \(a\tanh(kx)\) | \(ak\,\textrm{sech}^2(kx)\) |
| \(a\,\textrm{coth}(kx)\) | \(-ak\,\textrm{csch}^2(kx)\) |
| \(a\,\textrm{sech}(kx)\) | \(-ak\,\textrm{sech}(kx)\,\textrm{tanh}(kx)\) |
| \(a\,\textrm{csch}(kx)\) | \(-ak\,\textrm{csch}(kx)\,\textrm{coth}(kx)\) |
Derivatives of inverse hyperbolic functions
Throughout, \(a,k\) are real numbers.
| function | derivative w.r.t \(x\) | notes |
|---|---|---|
| \(a\sinh^{-1}(kx)\) | \(\dfrac{ak}{\sqrt{1+k^2x^2}}\) | |
| \(a\cosh^{-1}(kx)\) | \(\dfrac{ak}{\sqrt{k^2x^2 - 1}}\) | \(a,k,x\) positive |
| \(a\tanh^{-1}(kx)\) | \(\dfrac{ak}{1-k^2x^2}\) | |
| \(a\,\textrm{coth}^{-1}(kx)\) | \(\dfrac{ak}{1-k^2x^2}\) | |
| \(a\,\textrm{sech}^{-1}(kx)\) | \(-\dfrac{ak}{x\sqrt{1-k^2x^2}}\) | \(a,k,x\) positive |
| \(a\,\textrm{csch}^{-1}(kx)\) | \(-\dfrac{ak}{|x|\sqrt{k^2x^2+1}}\) |
Further reading
For more about where these came from, please see Guide: Introduction to differentiation and the derivative and [Proof sheet: Derivatives of other common functions].
Version history
v1.0: created in 08/25 by tdhc.