How to Find the Derivative: Power, Product, Quotient and Chain Rules

The derivative of a function tells you its rate of change: how fast $y$ changes as $x$ changes, or, on a graph, the slope of the curve at each point. Most derivatives you will meet in a course come from four rules. Spot which one the function needs, apply it, and simplify.

The calculator is loaded with example 2 from the power rule section. Type your own function using ^ for powers and * for multiplication, for example x^2*sin(x).

Which rule do I use?

If the function is	Example	Use
A power of x, or a sum of them	$3x^4 - 7x^2 + 5x$	Power rule, term by term
Two functions multiplied	$x^2 \sin x$	Product rule
One function divided by another	$\dfrac{x^2 + 1}{x - 3}$	Quotient rule
A function inside another function	$(3x^2 + 5)^4$	Chain rule

Many functions need two rules at once; there is an example of that further down.

What a derivative is

The derivative of $f(x)$ at a point is the slope of the tangent line there. You get it by taking the slope between two points on the curve and letting the gap $h$ between them shrink to zero:

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

y = x^2

, the slope from

x = 1

x = 1 + h

2 + h

. As

h

shrinks, the dashed lines close in on the tangent, whose slope is 2.

For $f(x) = x^2$ the limit works out like this:

\begin{aligned} \frac{(x + h)^2 - x^2}{h} &= \frac{2xh + h^2}{h} \\ &= 2x + h \\ &\to 2x \text{ as } h \to 0 \end{aligned}

So $f'(x) = 2x$ , and at $x = 1$ the slope is 2, as in the picture. Working from the limit every time would be slow, which is what the rules below are for.

Rule 1: The power rule

\frac{d}{dx} x^n = n x^{n-1}

Bring the power down in front and reduce the power by one. A constant on its own differentiates to zero, and a constant multiplier stays where it is.

Example 1

$f(x) = x^5$

f'(x) = 5x^4

Example 2: a polynomial

$f(x) = 3x^4 - 7x^2 + 5x - 9$

Differentiate term by term:

f'(x) = 12x^3 - 14x + 5

Example 3: negative and fractional powers

Rewrite roots and fractions as powers first: $\frac{1}{x^3} = x^{-3}$ and $\sqrt{x} = x^{1/2}$ .

\frac{d}{dx} x^{-3} = -3x^{-4} = -\frac{3}{x^4}

\frac{d}{dx} x^{1/2} = \frac{1}{2}x^{-1/2} = \frac{1}{2\sqrt{x}}

Rule 2: The product rule

\frac{d}{dx}[u v] = u' v + u v'

Why the rule has two terms: think of

uv

as the area of a rectangle. When

x

changes a little, the area grows by a strip

v\,du

and a strip

u\,dv

; the corner

du\,dv

is so small it vanishes in the limit.

Example 4

$f(x) = x^2 \sin x$ . Let $u = x^2$ and $v = \sin x$ , so $u' = 2x$ and $v' = \cos x$ .

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

Rule 3: The quotient rule

\frac{d}{dx}\left[\frac{u}{v}\right] = \frac{u' v - u v'}{v^2}

Example 5

$f(x) = \frac{x^2 + 1}{x - 3}$ . Let $u = x^2 + 1$ and $v = x - 3$ , so $u' = 2x$ and $v' = 1$ .

\begin{aligned} f'(x) &= \frac{2x(x - 3) - (x^2 + 1)}{(x - 3)^2} \\ &= \frac{x^2 - 6x - 1}{(x - 3)^2} \end{aligned}

Rule 4: The chain rule

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

Differentiate the outside function, leave the inside alone, then multiply by the derivative of the inside.

Example 6 as a pipeline:

x

feeds the inside function, whose output feeds the outside one. Each stage multiplies the rate of change, so the two derivatives multiply.

Example 6

$f(x) = (3x^2 + 5)^4$ . The outside is $u^4$ ; the inside is $u = 3x^2 + 5$ , with derivative $6x$ .

\begin{aligned} f'(x) &= 4(3x^2 + 5)^3 \cdot 6x \\ &= 24x(3x^2 + 5)^3 \end{aligned}

Example 7

$f(x) = \sin(5x^3)$

\begin{aligned} f'(x) &= \cos(5x^3) \cdot 15x^2 \\ &= 15x^2 \cos(5x^3) \end{aligned}

Using two rules at once

$f(x) = x^2 e^{3x}$ is a product, and its second factor needs the chain rule: $\frac{d}{dx} e^{3x} = 3e^{3x}$ .

\begin{aligned} f'(x) &= 2x e^{3x} + x^2 \cdot 3e^{3x} \\ &= x e^{3x}(3x + 2) \end{aligned}

What the derivative tells you

Where $f'(x)$ is positive the function is rising, where it is negative the function is falling, and where it is zero the graph is flat. These stationary points are where you look for maximums and minimums in optimisation problems. Most are turning points, but not all: $x^3$ is flat at 0 and keeps on rising.

f(x) = x^3 - 3x

(blue) and its derivative

f'(x) = 3x^2 - 3

(amber). The derivative is zero at

x = -1

and

x = 1

, exactly where

f

turns.

Derivatives to know by heart

$\frac{d}{dx}\left[c\right]$	$0$
$\frac{d}{dx}\left[x^n\right]$	$nx^{n-1}$
$\frac{d}{dx}\left[e^x\right]$	$e^x$
$\frac{d}{dx}\left[a^x\right]$	$a^x \ln a$
$\frac{d}{dx}\left[\ln x\right]$	$\frac{1}{x}$
$\frac{d}{dx}\left[\sin x\right]$	$\cos x$
$\frac{d}{dx}\left[\cos x\right]$	$-\sin x$
$\frac{d}{dx}\left[\tan x\right]$	$\sec^2 x$

Common mistakes

Forgetting the inside derivative. The derivative of $\sin(5x^3)$ is not $\cos(5x^3)$ ; it needs the extra factor $15x^2$ .
Getting the quotient rule backwards. The top is $u'v - uv'$ , in that order, because subtraction does not swap.
Treating a product like a sum. The derivative of $x^2 \sin x$ is not $2x \cos x$ .
Leaving roots as roots. Rewrite $\sqrt{x}$ as $x^{1/2}$ before using the power rule.

Related calculators and guides

Derivative calculator: first, higher-order and at-a-point derivatives with steps.
Partial derivative calculator for functions of several variables (guide).
Implicit differentiation calculator for equations like $x^2 + y^2 = 25$ (guide).
Limit calculator and how to calculate limits.
Going the other way: integral calculator and integrals and antiderivatives.