Differentiation: rules and techniques

Learning objectives

Understand the derivative as the gradient of the tangent and as a rate of change, and use the notations f′(x), f″(x), dy/dx and d²y/dx² correctly

Differentiate xⁿ for any rational n after rewriting roots and reciprocals in index form, together with sin x, cos x, tan x, eˣ and ln x

Apply the chain rule to differentiate composite functions such as (ax + b)ⁿ, sin(ax + b), e^(ax + b) and ln(ax + b)

Apply the product rule and the quotient rule, and decide when rewriting a quotient as a product with a negative index gives a shorter solution

Combine the rules to differentiate expressions built from several functions, and simplify the resulting derivative by factorising

Find the second derivative of a function and explain what its sign says about how the gradient is changing