Standard results of differentiation

This lesson is a reference list of the standard derivative formulas, together with the basic rules of differentiation and the limit definition of the derivative. These are the results you use again and again when differentiating functions.

Powers, roots, and reciprocals

$\frac{d}{dx}\left(x^n\right) = n\,x^{n-1}$

$\frac{d}{dx}(x) = 1, \qquad \frac{d}{dx}(1) = 0, \qquad \frac{d}{dx}(c) = 0$

$\frac{d}{dx}\left(x^2\right) = 2x$

$\frac{d}{dx}\left(\sqrt{x}\right) = \frac{1}{2\sqrt{x}}$

$\frac{d}{dx}\left(\frac{1}{x}\right) = -\frac{1}{x^2}$

$\frac{d}{dx}\left(\frac{1}{x^2}\right) = -\frac{2}{x^3}$

Exponential and logarithmic functions

$\frac{d}{dx}\left(a^x\right) = a^x \log_e a$

In particular, $\dfrac{d}{dx}\left(e^x\right) = e^x$, since $\log_e e = 1$.

$\frac{d}{dx}\left(\log x\right) = \frac{1}{x}$

Trigonometric functions

$\frac{d}{dx}(\sin x) = \cos x, \qquad \frac{d}{dx}(\cos x) = -\sin x$

$\frac{d}{dx}(\tan x) = \sec^2 x, \qquad \frac{d}{dx}(\cot x) = -\csc^2 x$

$\frac{d}{dx}(\sec x) = \sec x \tan x, \qquad \frac{d}{dx}(\csc x) = -\csc x \cot x$

Product and quotient rules

Product rule:

$\frac{d}{dx}(uv) = u\,\frac{dv}{dx} + v\,\frac{du}{dx}$

Quotient rule:

$\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v\,\dfrac{du}{dx} - u\,\dfrac{dv}{dx}}{v^2}$

Inverse trigonometric functions

$\frac{d}{dx}\left(\sin^{-1} x\right) = \frac{1}{\sqrt{1-x^2}}, \qquad \frac{d}{dx}\left(\cos^{-1} x\right) = -\frac{1}{\sqrt{1-x^2}}$

$\frac{d}{dx}\left(\tan^{-1} x\right) = \frac{1}{1+x^2}, \qquad \frac{d}{dx}\left(\cot^{-1} x\right) = -\frac{1}{1+x^2}$

$\frac{d}{dx}\left(\sec^{-1} x\right) = \frac{1}{x\sqrt{x^2-1}}, \qquad \frac{d}{dx}\left(\csc^{-1} x\right) = -\frac{1}{x\sqrt{x^2-1}}$

Rules of differentiation

Sum and difference rule:

$\frac{d}{dx}\big(f(x) \pm g(x)\big) = \frac{d}{dx}f(x) \pm \frac{d}{dx}g(x)$

Constant-multiple rule: for a constant $c$,

$\frac{d}{dx}\big(c\,f(x)\big) = c\,\frac{d}{dx}f(x)$

As an example,

$\frac{d}{dx}(ax + b) = \frac{d}{dx}(ax) + \frac{d}{dx}(b) = a + 0 = a.$

Derivative from first principles

The derivative is defined as a limit:

$\frac{dy}{dx} = f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}.$

Evaluated at a point $x = a$, this gives

$f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}.$

Key takeaways

• Memorise the power rule $\dfrac{d}{dx}(x^n) = n\,x^{n-1}$ and the derivatives of the exponential, logarithmic, trigonometric, and inverse trigonometric functions.
• Combine functions using the sum, constant-multiple, product, and quotient rules.
• Every one of these results follows from the first-principles limit $f'(x) = \lim_{h \to 0}\dfrac{f(x+h) - f(x)}{h}$.