Derivative Calculator

Calculate Derivative

Enter a function to differentiate with respect to x

Quick examples:

Differentiation Rules

Power Rule

$\frac{d}{dx}[x^n] = nx^{n-1}$

$\frac{d}{dx}[x^3] = 3x^2$

Constant Rule

$\frac{d}{dx}[c] = 0$

$\frac{d}{dx}[5] = 0$

Sum Rule

$\frac{d}{dx}[f + g] = f' + g'$

$\frac{d}{dx}[x^2 + x] = 2x + 1$

Product Rule

$\frac{d}{dx}[fg] = f'g + fg'$

$\frac{d}{dx}[x \sin x] = \sin x + x\cos x$

Quotient Rule

$\frac{d}{dx}[\frac{f}{g}] = \frac{f'g - fg'}{g^2}$

$\frac{d}{dx}[\frac{x}{x+1}] = \frac{1}{(x+1)^2}$

Chain Rule

$\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$

$\frac{d}{dx}[\sin(2x)] = 2\cos(2x)$

Common Derivatives

Trigonometric Functions

$\frac{d}{dx}[\sin x]$	$= \cos x$
$\frac{d}{dx}[\cos x]$	$= -\sin x$
$\frac{d}{dx}[\tan x]$	$= \sec^2 x$
$\frac{d}{dx}[\sec x]$	$= \sec x \tan x$
$\frac{d}{dx}[\csc x]$	$= -\csc x \cot x$
$\frac{d}{dx}[\cot x]$	$= -\csc^2 x$

Exponential & Logarithmic

$\frac{d}{dx}[e^x]$	$= e^x$
$\frac{d}{dx}[a^x]$	$= a^x \ln a$
$\frac{d}{dx}[\ln x]$	$= \frac{1}{x}$
$\frac{d}{dx}[\log_a x]$	$= \frac{1}{x \ln a}$

Inverse Trig Functions

$\frac{d}{dx}[\arcsin x]$	$= \frac{1}{\sqrt{1-x^2}}$
$\frac{d}{dx}[\arccos x]$	$= -\frac{1}{\sqrt{1-x^2}}$
$\frac{d}{dx}[\arctan x]$	$= \frac{1}{1+x^2}$

Worked Examples

Example 1: Power Rule

Find the derivative of $f(x) = 3x^4 - 2x^2 + 5x - 7$

Apply the power rule to each term:

$f'(x) = \frac{d}{dx}[3x^4] - \frac{d}{dx}[2x^2] + \frac{d}{dx}[5x] - \frac{d}{dx}[7]$

$f'(x) = 3 \cdot 4x^3 - 2 \cdot 2x + 5 \cdot 1 - 0$

$f'(x) = 12x^3 - 4x + 5$

Example 2: Product Rule

Find the derivative of $f(x) = x^2 \sin x$

Let $u = x^2$ and $v = \sin x$

Then $u' = 2x$ and $v' = \cos x$

Using product rule: $f'(x) = u'v + uv'$

$f'(x) = 2x \sin x + x^2 \cos x$

Example 3: Chain Rule

Find the derivative of $f(x) = (3x + 2)^5$

Let $u = 3x + 2$, then $f(x) = u^5$

$\frac{df}{du} = 5u^4$ and $\frac{du}{dx} = 3$

By chain rule: $\frac{df}{dx} = \frac{df}{du} \cdot \frac{du}{dx}$

$f'(x) = 5(3x + 2)^4 \cdot 3 = 15(3x + 2)^4$

Practice Problems

Find $\frac{d}{dx}[x^5 - 3x^3 + 2x]$
Find $\frac{d}{dx}[e^x \cos x]$
Find $\frac{d}{dx}[\frac{x^2}{x+1}]$
Find $\frac{d}{dx}[\ln(x^2 + 1)]$
Find $\frac{d}{dx}[\sin^2 x]$
Find $\frac{d}{dx}[\sqrt{x^2 + 4}]$

Show Answers

$5x^4 - 9x^2 + 2$
$e^x \cos x - e^x \sin x$
$\frac{x^2 + 2x}{(x+1)^2}$
$\frac{2x}{x^2 + 1}$
$2\sin x \cos x$
$\frac{x}{\sqrt{x^2 + 4}}$