The rules
Each name links to a page with five worked examples and a graph.
| Rule | Statement |
|---|---|
| Constant | \(\frac{d}{dx}c = 0\) |
| Variable | \(\frac{d}{dx}x = 1\) |
| Sum | \(\left(f + g\right)' = f' + g'\) |
| Constant multiple | \(\left(cf\right)' = c\,f'\) |
| Power | \(\frac{d}{dx}x^{n} = n x^{n-1}\) |
| Product | \(\left(fg\right)' = f'g + fg'\) |
| Quotient | \(\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^{2}}\) |
| Chain | \(\left(f \circ g\right)' = f'(g)\,g'\) |
| Exponential | \(\frac{d}{dx}a^{x} = a^{x}\ln a\) |
| Logarithm | \(\frac{d}{dx}\ln x = \frac{1}{x}\) |
| Logarithm, base a | \(\frac{d}{dx}\log_{a} x = \frac{1}{x\ln a}\) |
| Trigonometric | \(\frac{d}{dx}\sin x = \cos x\) |
| Inverse trig | \(\frac{d}{dx}\arctan x = \frac{1}{1 + x^{2}}\) |
| Hyperbolic | \(\frac{d}{dx}\tanh x = \operatorname{sech}^{2} x\) |
| Inverse hyperbolic | \(\frac{d}{dx}\operatorname{artanh} x = \frac{1}{1 - x^{2}}\) |
| Absolute value | \(\frac{d}{dx}\left|x\right| = \operatorname{sgn} x, \; x \neq 0\) |
| Logarithmic | \(\frac{d}{dx}f^{g} = f^{g}\left(g'\ln f + \frac{gf'}{f}\right)\) |
| Rewrite in natural logs | \(\log_{a} b = \frac{\ln b}{\ln a}\) |
Powers and roots
Exponential and logarithm
Trigonometric
| f(x) | f′(x) | Open |
|---|---|---|
| \(\sin\left(x\right)\) | \(\cos\left(x\right)\) | steps |
| \(\cos\left(x\right)\) | \(-\sin\left(x\right)\) | steps |
| \(\tan\left(x\right)\) | \(\sec^{2}\left(x\right)\) | steps |
| \(\sec\left(x\right)\) | \(\sec\left(x\right) \tan\left(x\right)\) | steps |
| \(\csc\left(x\right)\) | \(-\cot\left(x\right) \csc\left(x\right)\) | steps |
| \(\cot\left(x\right)\) | \(-\csc^{2}\left(x\right)\) | steps |
Inverse trigonometric
Hyperbolic
Inverse hyperbolic
Other
| f(x) | f′(x) | Open |
|---|---|---|
| \(\left|x\right|\) | \(\operatorname{sgn}\left(x\right)\) | steps |
Using the sheet
- The rules tell you how to take an expression apart; the tables tell you what the pieces differentiate to.
- Every standard derivative above assumes radians, and \(\ln\) means the natural logarithm.
- \(n\) and \(a\) are constants. Where the variable sits in the exponent, use the exponential rule, not the power rule.
- This page prints cleanly — the site's print stylesheet drops the navigation and keeps the tables.