Differentiation

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Cambridge International AS & A Level Mathematics 9709 — Pure Mathematics 3

Differentiation

The big idea: Learn six standard derivatives and three rules (chain, product, quotient), and you can differentiate any function in this paper, including curves given parametrically or implicitly.

Summary — What This Chapter Covers

d/dx(e^x) = e^x, d/dx(ln x) = 1/x, d/dx(sin x) = cos x, d/dx(cos x) = −sin x, d/dx(tan x) = sec²x, d/dx(tan⁻¹x) = 1/(1 + x²).
x must be in radians for the trig derivatives.
Chain rule for composites, product rule for uv, quotient rule for u/v.
Parametric: dy/dx = (dy/dt) ÷ (dx/dt).
Implicit: differentiate every term with respect to x; a term in y picks up a factor dy/dx.

1. Derivatives of e^x, ln x and Trigonometric Functions

Standard results

The six you must know

With the chain rule, each standard result extends to a linear (or any) inner function: d/dx e^f(x) = f′(x)e^f(x), d/dx ln f(x) = f′(x)/f(x), d/dx sin f(x) = f′(x)cos f(x). Derivatives of sin⁻¹x and cos⁻¹x are not required.

Standard derivatives e^{ax + b} → ae^{ax + b} ln(ax + b) → a/(ax + b) sin ax → a cos ax cos ax → −a sin ax
tan ax → a sec²ax tan⁻¹x → 1/(1 + x²)

Worked Example

Differentiate y = e^3x + ln(2x) + sin 4x + tan⁻¹(2x).

Simplify logs first

ln(2x) = ln 2 + ln x, so its derivative is just 1/x. Likewise ln(x³) = 3 ln x. Splitting the log before differentiating often saves a quotient rule.

2. Product, Quotient and Chain Rules

Rules

Building derivatives of combined functions

Identify the structure first: a function of a function (chain), a product of two functions (product), or one function over another (quotient). Many questions then ask for a stationary point: set dy/dx = 0. An exponential factor such as e^−2x is never zero, so it can be cancelled.

Rules Chain: dy/dx = (dy/du)(du/dx) Product: (uv)′ = u′v + uv′ Quotient: (u/v)′ = (u′v − uv′)/v²

Worked Example 1

Differentiate (a) x² ln x (b) xe^{1 − x²} (c) (2x − 4)/(3x + 2).

Worked Example 2

Find the exact coordinates of the stationary point on y = xe^−2x.

Practice Question

Find the x-coordinate of the stationary point on y = (ln x)/x and state the exact value of y there.

3. Parametric and Implicit Differentiation

Parametric & implicit

Gradients without y = f(x)

Parametric: when x and y are both given in terms of t, find dx/dt and dy/dt and divide. To find a tangent or normal at a point, find the value of t first.

Implicit: differentiate both sides with respect to x. d/dx(y²) = 2y dy/dx, and a product such as xy needs the product rule: d/dx(xy) = y + x dy/dx. Then collect the dy/dx terms.

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Differentiation