G: Differentiation: Differentiate xⁿ, for rational values of n, and related constant multiples, sums and differences. Differentiate e^(kx) and a^(kx), sin kx, cos kx, tan kx and related sums, differences and constant multiples. Understand and use the derivative of ln x as 1/x.

The idea

This topic covers the standard derivatives required at A-level. For powers, use d/dx(x^n)=n x^{n-1} for any rational n, including negative and fractional indices. Apply the constant multiple, sum and difference rules term by term. For exponentials, d/dx(e^{kx})=k e^{kx} and d/dx(a^{kx})=k a^{kx} ln a. For trigonometric functions, d/dx(sin kx)=k cos kx, d/dx(cos kx)=-k sin kx, and d/dx(tan kx)=k sec^2 kx. Combine these with sums, differences and constant multiples. Also know that d/dx(ln x)=1/x. For example, differentiate 3x^{1/2} - 2e^{3x} + 5 sin 2x: the derivative is (3/2)x^{-1/2} - 6e^{3x} + 10 cos 2x. This skill underpins later calculus work.

What earns the marks

• Apply the power rule d/dx(x^n)=n x^{n-1} correctly for rational n, including negative and fractional indices.
• Differentiate constant multiples, sums and differences term by term, preserving coefficients.
• Differentiate e^{kx} as k e^{kx} and a^{kx} as k a^{kx} ln a.
• Differentiate sin kx, cos kx and tan kx as k cos kx, -k sin kx and k sec^2 kx respectively.
• State and use the derivative of ln x as 1/x, recognising that ln x is not differentiated by the power rule.

Common mistakes and how to put them right

• Forgetting to multiply by the constant k when differentiating e^{kx}, sin kx, etc.

  Put it right: Apply the chain rule factor k.

• Treating ln x as a power of x, e.g. rewriting ln x as x^{-1} and differentiating to -x^{-2}.

  Put it right: d/dx(ln x)=1/x; ln x is not a power function, so the power rule does not apply.

• Omitting the constant k when differentiating tan kx, e.g. writing sec^2 kx instead of k sec^2 kx.

  Put it right: d/dx(tan kx)=k sec^2 kx, so the factor k must be included.

Exam tips

• Write down the standard derivative you are using before applying it, especially for trigonometric and exponential functions.
• Check each term separately; do not rush and miss constant multiples.
• For expressions with several terms, differentiate term by term and simplify where possible.