Differentiation

Sections

1. 10A Tangents and the derivative The derivative as the gradient of the tangent, sketching the gradient function, derivatives of linear and constant functions and semicircles, and secants versus tangents.
2. 10B The derivative as a limit The tangent as the limit of secants, differentiating from first principles, finding points with a given gradient, and where a function is not differentiable.
3. 10C A rule for differentiating powers of x The power rule, differentiating sums and multiples, expanding products first, and finding tangents and normals to a curve.
4. 10D The notation dy/dx for the derivative Small changes Δx and Δy, Leibniz notation dy/dx, operator notation d/dx, and second and higher derivatives.
5. 10E The chain rule Composite functions, the chain rule dy/dx = dy/du × du/dx, powers of a linear function, and tangents using the chain rule.
6. 10F Differentiating powers with negative indices Extending the power rule to negative indices, rewriting fractions as powers, and combining negative indices with the chain rule.
7. 10G Differentiating powers with fractional indices Differentiating square roots, cube roots and other fractional powers, using the chain rule with roots, and where these derivatives are undefined.
8. 10H The product rule Differentiating a product of two functions with (uv)′ = u′v + uv′, combining it with the chain rule, and factorising the result.
9. 10I The quotient rule Differentiating a quotient with (u/v)′ = (u′v − uv′)/v², simplifying the result, and choosing between the quotient rule and rewriting.
10. 10J Rates of change Average and instantaneous rates of change, the derivative as a rate, and where a function is increasing, decreasing or stationary.
11. 10K Average velocity and average speed Displacement as a function of time, graphing motion, average velocity, distance travelled, and average speed.
12. 10L Instantaneous velocity and speed Velocity as the derivative of displacement, speed as its size, vectors and scalars, and finding when a particle is stationary.