AS Level Mathematics · Pure Mathematics 1. A short explanation of the idea, the rules to remember, the mistake to avoid, a worked example and practice questions with answers.
The derivative dy/dx is the gradient of a curve at a point. It is the limit of the gradients of chords as the two ends of the chord move closer together. It also measures how fast y changes when x changes.
Before you differentiate, write every term as a power of x. For a bracket raised to a power, use the chain rule: differentiate the outside, then multiply by the derivative of the inside.
The gradient then answers most questions. It gives the tangent and the normal at a point. Its sign shows where a function is increasing or decreasing. Where it is zero there is a stationary point, and the second derivative tells you if that point is a maximum or a minimum.
Students forget to multiply by the derivative of the bracket in the chain rule. The derivative of (2x − 3)5 is 5(2x − 3)4 × 2 = 10(2x − 3)4, not 5(2x − 3)4.
Answer: x + 12y = 13
Try each one, then open the answer.
Answer: B. dy/dx = 2x − 3 = 3 at x = 3, so the normal gradient is −1/3. y − 0 = −(1/3)(x − 3) gives 3y = −x + 3, i.e. x + 3y = 3. y = 3x − 9 is the tangent.
Answer: A. y = 4(3x − 1)−2, so dy/dx = 4 × (−2)(3x − 1)−3 × 3 = −24/(3x − 1)3. Both the −2 and the chain-rule factor 3 are needed.
Answer: C. f′(x) = 6x2 − 2x and f″(x) = 12x − 2, so f″(1) = 10. Differentiate twice, not once.
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