# Differentiation

AS Level

• 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
• Understand and use the second derivative as the rate of change of gradient
• Differentiate xn, for rational values of n, and related constant multiples, sums and differences
• Apply differentiation to find gradients, tangents and normals, maxima and minima and stationary points
• Identify where functions are increasing or decreasing

A Level

• connection to convex and concave sections of curves and points of inflection
• Differentiate ekx and akx, sinkx, coskx ,tankx and related sums, differences and constant multiples
• Understand and use the derivative of lnx
• points of inflection
• Differentiate using the product rule, the 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, (contexts may include kinematics, population growth and modelling the relationship between price and demand)