Differentiation, tangents and normals

What you’ll learn

• Differentiate sums of rational powers.
• Interpret a derivative as gradient and rate of change.
• Find tangent and normal equations.

Understand the idea

A derivative gives the instantaneous rate of change and the gradient of the tangent. It arises as the limit of a secant gradient as the two points approach each other. Differentiate a power by multiplying by its exponent and reducing the exponent by one; constants differentiate to zero.

Expand products and rewrite roots or quotients as powers before using the power rule in P1. Differentiate again to obtain the second derivative. To find a tangent, calculate both the point on the curve and the gradient there, then use the straight-line equation. The normal is perpendicular to the tangent.

Key formulae

$$f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}h$$

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

$$y-y_0=f'(x_0)(x-x_0)$$

$$m_{\mathrm{normal}}=-\frac1{f'(x_0)}\quad(f'(x_0)\ne0)$$

Worked example

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

1. The point has $y=8-4=4$, so it is $(2,4)$.
2. Differentiate: $dy/dx=3x^2-2$. At $x=2$ the gradient is $10$.
3. Substitute into point-gradient form: $y-4=10(x-2)$.

Answer: $y=10x-16$.