Differentiating axⁿ; tangents and normals

Maths: Applications and Interpretation · 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

To differentiate a power of x, multiply by the power and then lower the power by one. Do this to each term in turn. A constant term disappears, because its graph is flat. Before you start, rewrite every term as a power of x: for example 3/x² becomes 3x⁻², and a fraction such as (x² + 1)/x is split into x + x⁻¹.

Putting an x-value into dy/dx gives the gradient of the curve at that point. The tangent is the straight line that touches the curve there with the same gradient. The normal is the line through the same point at right angles to the tangent.

Rules to remember

If y = axⁿ, then dy/dx = anx^{n − 1}
The derivative of ax is a; the derivative of a constant is 0
Negative powers follow the same rule: x⁻² differentiates to −2x⁻³
Gradient at a point: put the x-value into dy/dx
Tangent at (x₁, y₁) with gradient m: y − y₁ = m(x − x₁)
Gradient of the normal = −1 ÷ m, because the two gradients multiply to −1
Find y₁ from the ORIGINAL equation of the curve, not from dy/dx

Common mistake

With a negative power students raise the power instead of lowering it: the derivative of x⁻² is −2x⁻³, not −2x⁻¹. For a normal, remember to change the gradient to −1 ÷ m before writing the line.

Worked example

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

Point: y = 2³ − 2 × 2 = 8 − 4 = 4, so the point is (2, 4)
dy/dx = 3x² − 2; at x = 2 the gradient is 3 × 4 − 2 = 10
Tangent: y − 4 = 10(x − 2), so y = 10x − 16
Normal: gradient = −1 ÷ 10 = −0.1, so y − 4 = −0.1(x − 2), giving y = −0.1x + 4.2

Answer: Tangent: y = 10x − 16. Normal: y = −0.1x + 4.2.

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Topic 5: Calculus

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