Differentiate y = e2x sin x.
- This is a product: u = e2x and v = sin x
- du/dx = 2e2x (chain rule) and dv/dx = cos x
- dy/dx = u(dv/dx) + v(du/dx) = e2x cos x + 2e2x sin x
- Factorise: dy/dx = e2x(cos x + 2 sin x)
Answer: dy/dx = e2x(cos x + 2 sin x)
Further differentiation
Maths: Analysis and Approaches · Calculus. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.
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The idea
This topic adds the standard derivatives of sin x, cos x, ex and ln x, and powers of x with fractional or negative indices. Angles must be in radians for the trigonometric results to hold.
Three rules deal with functions built from simpler ones. The chain rule is for a function inside another function, such as (2x + 1)5 or cos 4x: differentiate the outside, keep the inside unchanged, then multiply by the derivative of the inside. The product rule is for two functions multiplied together, and the quotient rule is for one function divided by another.
First decide which structure you have. Many questions need two rules together.
Rules to remember
- d/dx(xn) = nxn − 1 for any rational n, so d/dx(√x) = 1/(2√x)
- d/dx(sin x) = cos x; d/dx(cos x) = −sin x
- d/dx(ex) = ex; d/dx(ln x) = 1/x
- Chain rule: dy/dx = (dy/du) × (du/dx)
- So d/dx(eu) = u′eu, d/dx(ln u) = u′/u, d/dx(sin u) = u′cos u, d/dx(cos u) = −u′sin u
- Product rule: if y = uv then dy/dx = u(dv/dx) + v(du/dx)
- Quotient rule: if y = u/v then dy/dx = [v(du/dx) − u(dv/dx)]/v2
Common mistake
Forgetting to multiply by the derivative of the inside function. The derivative of (2x + 1)5 is 10(2x + 1)4, not 5(2x + 1)4.
Worked example
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