Differentiation and Applications
Core ideas
Candidates must differentiate powers of x, polynomials, and simple composite functions using the chain rule; know that the derivative is the gradient of the tangent; and apply differentiation to gradients, tangents and normals, increasing and decreasing functions, and maxima and minima. The product and quotient rules may appear. Second derivatives distinguish maximum from minimum: if d²y/dx² < 0 it is a maximum, if > 0 a minimum. Rates of change and small increments also feature, using δy ≈ (dy/dx)δx.
Traps
- Forgetting the chain rule factor in expressions like (2x+3)⁵.
- Using the tangent gradient for the normal (must be negative reciprocal).
- Not verifying the nature of a stationary point with the second derivative or a sign test.
- Differentiating constants incorrectly (derivative of a constant is zero).
- Confusing dy/dx with the equation of the tangent.
Worked example
Find the stationary points of y = x³ − 3x. dy/dx = 3x² − 3 = 0, so x = ±1. Then y = −2 at x = 1 and y = 2 at x = −1. Second derivative 6x: at x = 1, positive, minimum; at x = −1, negative, maximum.
Rule of thumb
Every stationary-point question should end with a nature check - second derivative or a table of signs - before you state the answer.