Introduction to differentiation, tangents and normals

Maths: Analysis and Approaches · Calculus. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

The derivative f′(x), also written dy/dx, gives the gradient of a curve at any point. It comes from a limit: take a chord between two points on the curve, let the second point move towards the first, and the gradient of the chord approaches the gradient of the tangent.

The derivative is also a rate of change. dV/dt = −3 means V is falling by 3 units for each unit of time at that moment.

Where f′(x) is positive the function is increasing, and where it is negative the function is decreasing. The tangent at a point has the same gradient as the curve there, and the normal is perpendicular to the tangent.

Rules to remember

If y = axⁿ then dy/dx = anx^{n − 1}; differentiate a sum term by term
The derivative of a constant is 0; the derivative of ax is a
Rewrite first: 1/xⁿ = x⁻ⁿ
Increasing where f′(x) > 0, decreasing where f′(x) < 0, stationary where f′(x) = 0
Tangent at (x₁, y₁): y − y₁ = m(x − x₁), where m = f′(x₁)
Normal: same point, gradient −1/m
Parallel lines have equal gradients

Common mistake

Using the tangent gradient for the normal, or only changing its sign. The normal gradient is the negative reciprocal: if the tangent gradient is 10, the normal gradient is −1/10.

Worked example

Find the equations of the tangent and the normal to y = x³ − 2x at x = 2.

At x = 2, y = 2³ − 2 × 2 = 8 − 4 = 4, so the point is (2, 4)
dy/dx = 3x² − 2, so at x = 2 the gradient is 3 × 4 − 2 = 10
Tangent: y − 4 = 10(x − 2), so y = 10x − 16
Normal: gradient −1/10, so y − 4 = −(1/10)(x − 2), which gives x + 10y − 42 = 0

Answer: Tangent y = 10x − 16; normal x + 10y − 42 = 0

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Topic 5: Calculus

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