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 a rate of change of gradient; connection to convex and concave sections of curves and points of inflection.
Differentiate x^n for rational values of n, and related constant multiples, sums and differences;
Differentiate e^kx, a^kx, sin kx, cos kx and tan kx and related constant multiples, sums and differences;
Understand and use the derivative of ln x;
Apply differentiation to find gradients, tangents and normals, maxima and minima and stationary points and points of inflection;
Identify where functions area increasing or decreasing;
Differentiate using the product rule, quotient rule and the chain rule, including problems involving connected rates of change and inverse functions;
Differentiate simple functions and relations defined implicitly or parametrically, for first derivative only;
Construct simple differential equations in pure mathematics and in context, which may include kinematics, population growth and modelling the relationship between price and demand.
