The chain rule

You need the chain rule when one function sits inside another, e.g. (3x + 1)⁵ or e^(x²). It is the rule students forget most often in exams.

Rules

Three problems on this topic. Type like on a calculator: 3x^2 - 2/x, e^(2x), sqrt(x), ln(x).

Task 1 of 3Chain rule

Task

Differentiate:

f (x) = (5 x + 3)^{5}

$f^{'} (x) =$

Write like on a calculator: 3x^2 - 2/x, e^(2x), sqrt(x), ln(x), sin(x). Use brackets: e^(2x), not e^2x. Numbers also as 3/4 or e^2 - 1.

Worked example

Differentiate: $f (x) = (- 3 x + 3)^{5}$

Worked solution

1.Chain rule: outer derivative times inner derivative, $f^{'} (x) = g^{'} (h (x)) \cdot h^{'} (x)$ .
2.Outer: $g (z) = z^{5}$ , $g^{'} (z) = 5 z^{4}$ · Inner: $h (x) = - 3 x + 3$ , $h^{'} (x) = - 3$
3. $f^{'} (x) = 5 (- 3 x + 3)^{4} \cdot (- 3) = - 15 (- 3 x + 3)^{4}$

Typical mistake: Forgetting the inner derivative; then the factor $- 3$ is missing.

Differentiate: $f (x) = ln (x^{2} + 3)$

Worked solution

Typical mistake: Writing only $\frac{1}{h ( x )}$ and forgetting the inner derivative in the numerator.

Common questions

When a whole expression replaces x: in a bracket raised to a power, in the exponent of e, inside ln or inside sin.

Forgetting the inner derivative. For (3x + 1)⁵ the factor 3 is then missing.