10E

The chain rule

Composite functions, the chain rule dy/dx = dy/du × du/dx, powers of a linear function, and tangents using the chain rule.

$y = (3x - 1)^4$ is a composite: an inside function $u = 3x - 1$, then an outside function $y = u^4$.

Chain rule

$$\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$$

Example 1●

Differentiate $y = (3x - 1)^4$.

$\dfrac{dy}{dx} = 4(3x - 1)^3 \times 3 = 12(3x - 1)^3$

Example 2●●

Differentiate $y = (x^2 + 1)^5$.

$\dfrac{dy}{dx} = 5(x^2 + 1)^4 \times 2x = 10x(x^2 + 1)^4$

Inside u = x² + 1, so du/dx = 2x.

Example 3: a tangent●●●

Find the tangent to $y = (2x - 3)^3$ at $x = 2$.

At $x = 2$, $y = 1^3 = 1$.

$\dfrac{dy}{dx} = 6(2x - 3)^2 = 6$ at $x = 2$.

$y - 1 = 6(x - 2)$, so $y = 6x - 11$.

Tangents to y = (2x − 3)³

Blue dashed: the tangent at x = a. The gradient is never negative, and it is zero at x = 1.5.

Try this

Why is the tangent horizontal at $x = 1.5$?

Flashcards

Chain rule

$\dfrac{dy}{dx} = \dfrac{dy}{du} \times \dfrac{du}{dx}$

$rac{d}{dx}(ax + b)^n$

$an(ax + b)^{n-1}$

$rac{d}{dx}(x^2 + 1)^5$

$10x(x^2 + 1)^4$

Practice