Understand and use the derivative of \(f(x)\) as the gradient of the tangent to the graph of \(y = f(x)\) at a general point \((x,y)\); the gradient of the tangent as a limit; interpretation as a rate of change; sketching the gradient function for a given curve; second derivatives; differentiation from first principles for small positive integer powers of \(x\) and for \(\sin{x}\) and \(\cos{x}\).

Understand and use the second derivative as the rate of change of gradient; connection to convex and concave sections of curves and points of inflection.

Differentiation Differentiation from first principles Turning points

Differentiate \(x^n\), 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 \(\ln{x}\).

Differentiation of standard functions

Apply differentiation to find gradients, tangents and normals, maxima and minima and stationary points, points of inflection.

Identify where functions are increasing or decreasing.

Gradients, tangents and normals Turning points

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

Chain rule Connected rates of change Product rule Quotient rule

Differentiate simple functions and relations defined implicitly or parametrically, for first derivative only.

Implicit differentiation Parametric differentiation

Construct simple differential equations in pure mathematics and in context, (contexts may include kinematics, population growth and modelling the relationship between price and demand).

Differential equations