G: Differentiation: Differentiate using the product rule, the quotient rule and the chain rule, including problems involving connected rates of change and inverse functions.

The idea

This topic extends differentiation beyond simple polynomials. The product rule differentiates y = uv as u dv/dx + v du/dx. The quotient rule differentiates y = u/v as (v du/dx − u dv/dx)/v², where du/dx is the derivative of the numerator u and dv/dx is the derivative of the denominator v. The chain rule differentiates composite functions: dy/dx = dy/du × du/dx. For connected rates, use dy/dt = dy/dx × dx/dt, often with geometry formulae. For inverse functions, dy/dx = 1/(dx/dy), provided dx/dy ≠ 0. For example, differentiate y = x² sin x using the product rule, y = ln x / x using the quotient rule, and y = (3x + 1)^5 using the chain rule. Then solve a related-rates problem such as an expanding circle, and differentiate an inverse by finding dx/dy first.

What earns the marks

• Correctly identify the structure of the function: product, quotient, composite, connected rates or inverse, before choosing a rule.
• Apply the product rule accurately: for y = uv, dy/dx = u dv/dx + v du/dx, with correct derivatives of u and v.
• Apply the chain rule accurately: for y = f(g(x)), dy/dx = f'(g(x)) × g'(x), or in Leibniz notation dy/dx = dy/du × du/dx.
• For connected rates of change, set up a chain such as dV/dt = dV/dr × dr/dt and substitute known rates and dimensions correctly.
• For inverse functions, use dy/dx = 1/(dx/dy), ensuring the derivative is evaluated at the corresponding point and that dx/dy is not zero.

Common mistakes and how to put them right

• Writing the product rule as u'v' instead of u'v + uv'.

  Put it right: The sum of two terms, each with one differentiated factor.

• Reversing the numerator in the quotient rule to u dv/dx − v du/dx.

  Put it right: The correct numerator is v du/dx − u dv/dx, so the derivative of the numerator u (du/dx) appears first with a positive sign, and the derivative of the denominator v (dv/dx) appears second with a negative sign.

• Forgetting to multiply by the inner derivative in the chain rule, e.g. differentiating sin(2x) as cos(2x).

  Put it right: Multiply by the derivative of the inside function, giving 2cos(2x).

• In connected rates, omitting a necessary rate or using the wrong variable.

  Put it right: Write the full chain explicitly before substituting.

• For inverse functions, finding dx/dy and forgetting to take the reciprocal.

  Put it right: , the required derivative is dy/dx = 1/(dx/dy), so after differentiating x with respect to y, take the reciprocal and evaluate at the corresponding point.

Exam tips

• Before differentiating, classify the expression: is it a product, a quotient, a composition, or a combination? This determines the first rule to use.
• When using the chain rule with a function inside a function, write down the inner function u and the outer function y in terms of u to avoid missing factors.
• For connected rates, draw a diagram and list all given rates and quantities; then form the chain rule equation linking them.
• For inverse functions, find dx/dy first, then take the reciprocal, and substitute the y-value that corresponds to the given x-value.
• Check your answer by differentiating a simpler similar expression or by using numerical approximation for a specific value.