Topic 3 of 7

Differentiation rules and the chain rule

Step 15 · Calculus 1 topics

Differentiate polynomials, products, quotients and compositions with the standard rules, including exponential, logarithmic and trigonometric functions.

Power, sum and constant rules

You rarely need the limit definition: a few rules handle almost every function. Constants come along for the ride, and sums are differentiated term by term.

Function	Derivative
c	0
$x^{n}$	$nx^{n-1}$
$e^{x}$	$e^{x}$
$\ln x$	$\frac{1}{x}$
$\sin x$	$\cos x$
$\cos x$	$-\sin x$

Example: A polynomial

Differentiate f(x) = 4x³ − 5x² + 7x − 2.

Use the power rule on each term: 12x² − 10x + 7.
The constant −2 has derivative 0.

Answerf′(x) = 12x² − 10x + 7

Example: Roots and reciprocals

Differentiate $f(x) = \sqrt{x} + \frac{2}{x}$ .

Rewrite it as $x^{1/2} + 2x^{-1}$ .
Power rule: $\frac{1}{2}x^{-1/2} - 2x^{-2}$ .

Answer $f'(x) = \frac{1}{2\sqrt{x}} - \frac{2}{x^{2}}$

Product and quotient rules

(fg)' = f'g + fg' \qquad \left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^{2}}

Example: Product

Differentiate x² sin x.

Take f = x² and g = sin x, so f′ = 2x and g′ = cos x.
f′g + fg′ gives 2x sin x + x² cos x.

Answer2x sin x + x² cos x

Example: Quotient

Differentiate $\frac{x}{x^{2} + 1}$ .

Take f = x and g = x² + 1, so f′ = 1 and g′ = 2x.
$\frac{1 \cdot (x^{2} + 1) - x \cdot 2x}{(x^{2} + 1)^{2}}$ .

Answer $\frac{1 - x^{2}}{(x^{2} + 1)^{2}}$

The chain rule

For a function inside another function, f(g(x)), differentiate the outside, leave the inside alone, then multiply by the derivative of the inside.

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

Example: A power of a polynomial

Differentiate (3x² + 1)⁵.

Outside: u⁵ has derivative 5u⁴.
Inside: 3x² + 1 has derivative 6x.
Multiply: 5(3x² + 1)⁴ · 6x.

Answer30x(3x² + 1)⁴

Example: Exponential and trig

Differentiate $e^{\sin x}$ and cos(4x).

$e^{u}$ keeps its form, times the inside's derivative, cos x.
cos u becomes −sin u, times the inside's derivative, 4.

Answer $\cos x \, e^{\sin x}$ and −4 sin(4x)

Example: Logarithm

Differentiate ln(x² + 5).

The derivative of ln u is $\frac{1}{u}$ , times u′ = 2x.

Answer $\frac{2x}{x^{2} + 5}$

Try it yourself

Work each one on paper first, then open the answer to check.

Differentiate 6x⁴ − 3x + 9.
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24x³ − 3
Differentiate $x^{3}e^{x}$ .
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$3x^{2}e^{x} + x^{3}e^{x}$
Differentiate $\frac{\sin x}{x}$ .
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$\frac{x\cos x - \sin x}{x^{2}}$
Differentiate (2x − 7)⁴.
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8(2x − 7)³
Differentiate $e^{3x^{2}}$ .
Show answer
$6x\,e^{3x^{2}}$
Differentiate ln(5x).
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$\frac{1}{x}$
Find the slope of y = x³ − 4x at x = 2.
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