1.7 Differentiation

Understand the gradient of a curve at a point as the limit of the gradients of a suitable sequence of chords, and use the notations f′(x), f″(x), dy/dx and d²y/dx² for first and second derivatives.

Only an informal understanding of the idea of a limit is expected. Includes, for example, considering the gradient of the chord joining the points with x-coordinates 2 and (2 + h) on the curve y = x³. Formal use of the general method of differentiation from first principles is not required.

Taught in: The gradient of a curve, Differentiating powers of x · 8 questions

Use the derivative of xⁿ (for any rational n), together with constant multiples, sums and differences of functions, and of composite functions using the chain rule.

Example: find dy/dx given y = √(2x³ + 5).

Taught in: The chain rule, Differentiating powers of x · 12 questions

Apply differentiation to gradients, tangents and normals, increasing and decreasing functions, and rates of change.

Includes connected rates of change, e.g. given the rate of increase of the radius of a circle, find the rate of increase of its area for a specific value of one of the variables.

Taught in: Rates of change and connected rates, Increasing and decreasing functions, Tangents and normals · 17 questions

Locate stationary points and determine their nature, and use information about stationary points in sketching graphs.

Includes using the second derivative to identify maxima and minima; other methods may be used where a question does not specify one. Knowledge of points of inflexion is not included.

Taught in: Sketching curves with stationary points, Maximum and minimum problems, Stationary points and their nature · 19 questions