(A Level) Further differentiation

Learning objectives

Differentiate eˣ, ln x, sin x, cos x, tan x and tan⁻¹ x, together with constant multiples, sums and composites such as e^(3x²) or ln(cos 2x)

Use the product rule and the quotient rule for functions such as x³ ln x, e^(2x) cos x or (3x + 1)/(x − 2)²

Differentiate an implicitly defined curve such as x² + 3xy − y³ = 5 term by term, using the product rule for xy and d/dx[f(y)] = f′(y) dy/dx

Find dy/dx for a curve defined parametrically by x = f(t) and y = g(t) as (dy/dt) ÷ (dx/dt)

Apply these derivatives to find gradients, tangents, normals and stationary points, including points where the tangent is parallel to an axis