Differentiation
Connect tangent gradients and chain rules to stationary-point tests, numerical methods and constrained optimisation.
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Before you begin
Connect tangent gradients and chain rules to stationary-point tests, numerical methods and constrained optimisation.
Learning goals
- Interpret a derivative as gradient or rate of change
- Differentiate within the permitted function scope
- Differentiate power sums on the stated input interval
- Differentiate exponential compositions on the stated input interval
- Differentiate logarithmic compositions on the stated input interval
- Differentiate mixed function sums on the stated input interval
- Find a complete second derivative on the stated input interval
- Find the complete tangent to a polynomial at a stated input
- Apply the chain rule to composite functions
- Find and interpret a tangent to a curve
- Find and classify every stationary point of a polynomial
- Find the greatest polynomial value and every attaining input on a closed interval
- Find the least polynomial value and every attaining input on a closed interval
- Classify stationary points using justified derivative tests
- Model and solve an optimisation problem
Syllabus statements covered
- Include: derivative of f(x) as the gradient of the tangent to y = f(x) at a point; derivatives of x^n for any rational n, e^x and ln x, together with constant multiples, sums and differences; chain rule; graphical interpretation of f′(x) > 0, f′(x) = 0 and f′(x) < 0; first or second derivative tests for local maximum and minimum points and stationary points of inflexion in simple cases; locating maxima and minima and approximating a derivative at a point using a graphing calculator or graphing software; tangents to curves; local maxima and minima problems. Exclude: first-principles derivatives; products and quotients; dy/dx = 1/(dx/dy); implicit or parametric differentiation; non-stationary points of inflexion; connecting the graph of f to f′.
- Include: derivative of f(x) as the gradient of the tangent to y = f(x) at a point; derivatives of x^n for any rational n, e^x and ln x, together with constant multiples, sums and differences; chain rule; graphical interpretation of f′(x) > 0, f′(x) = 0 and f′(x) < 0; first or second derivative tests for local maximum and minimum points and stationary points of inflexion in simple cases; locating maxima and minima and approximating a derivative at a point using a graphing calculator or graphing software; tangents to curves; local maxima and minima problems. Exclude: first-principles derivatives; products and quotients; dy/dx = 1/(dx/dy); implicit or parametric differentiation; non-stationary points of inflexion; connecting the graph of f to f′.
Lessons
Work through them in order.
About lesson progress
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- Derivatives and the chain ruleDistinguish height from gradient, find first and second derivatives with their input conditions, and recover tangents from point or slope conditions.
- Stationary values and optimisationClassify stationary points, resolve inconclusive tests and compare feasible candidates, endpoints and ties in optimisation.