14. Differentiation and integration

• understand the idea of a derived functions
• use the derivatives of the standard functions xn (for any rational n), sinx, cos x, tan x, ex, ln x, together with constant multiples, sums and composite functions of these
• differentiate products and quotients of functions
• apply differentiation to gradients, tangents and normals, stationary points, connected rates of change, small increments and approximations and practical maxima and minima problems
• use the first and second derivative tests to discriminate between maxima and minima
• understand integration as the reverse process of differentiation
• integrate sums of terms in powers of x
• integrate functions of the form (ax + b)n for any rational n, sin (ax + b), cos (ax + b), e ax + b
• evaluate definite integrals and apply integration to the evaluation of plane areas
• apply differentiation and integration to kinematics problems that involve displacement, velocity and acceleration of a particle moving in a straight line with variable or constant acceleration, and the use of x–t
and v–t graphs