10C

A rule for differentiating powers of x

The power rule, differentiating sums and multiples, expanding products first, and finding tangents and normals to a curve.

The power rule

Power rule

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

Constant multiples and sums differentiate term by term: if $f(x) = ag(x) + bh(x)$, then $f'(x) = ag'(x) + bh'(x)$.

Example 1●

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

$f'(x) = 12x^2 - 10x + 7$

Example 2: expand first●●

Differentiate $y = (2x + 1)(x - 3)$.

$y = 2x^2 - 5x - 3$

$\dfrac{dy}{dx} = 4x - 5$

Example 3●●●

Find the tangent and normal to $y = x^2 - 4x + 5$ at $x = 3$.

$y = 9 - 12 + 5 = 2$, so the point is $(3, 2)$.

$\dfrac{dy}{dx} = 2x - 4 = 2$ at $x = 3$.

Tangent: $y - 2 = 2(x - 3)$, so $y = 2x - 4$.

Normal: gradient $-\tfrac12$, $y - 2 = -\tfrac12(x - 3)$, so $x + 2y - 7 = 0$.

Tangent and normal

Blue dashed: the tangent. Pink dashed: the normal, at right angles to it.

Try this

At which point is the normal vertical?

Flashcards

$rac{d}{dx},x^n$

$nx^{n-1}$

Tangent gradient

$f'(x_1)$

Normal gradient

$-\dfrac{1}{f'(x_1)}$

Practice