The chain rule

Differentiation

A-level Maths (9MA0)

ORIGINAL

Differentiate a 'function inside a function' — name the inside, then multiply the derivatives.

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Question

Differentiate

y = (3x^2 + 1)^5

with respect to

x

Every step worked, with the reasoning.

1
Let $u = 3x^2 + 1$ , so $y = u^5$ .
Name the inner function so it becomes a simple power.
2
$\dfrac{dy}{du} = 5u^4$ and $\dfrac{du}{dx} = 6x$
Differentiate the outside (power rule) and the inside separately.
3
$\dfrac{dy}{dx} = \dfrac{dy}{du} \times \dfrac{du}{dx} = 5u^4 \times 6x$
Chain rule: multiply the two derivatives together.
4
$\dfrac{dy}{dx} = 30x(3x^2 + 1)^4$
Substitute $u$ back and tidy.

Answer: $\dfrac{dy}{dx} = 30x(3x^2 + 1)^4$