Differentiation

Learn, then practise

The original worked lessons remain the main explanations. Use these focused reminders to check the conditions before you start.

Derivatives, chain rule and domains

Differentiate rational powers, sums and simplified expressions.
Use the chain rule with linear and nonlinear inner functions.
Connect a chord gradient to an instantaneous gradient; retain domains.

Write roots and reciprocals as powers. Use \(\displaystyle \frac{d}{dx}x^n=nx^{n-1}\) and \(\displaystyle \frac{d}{dx}[u(x)]^n=n[u(x)]^{n-1}u^{\prime}(x)\). Expand or simplify products and quotients first. A derivative is a gradient, not a coordinate; check where it is finite.

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Tangents, normals and complete line equations

Find the point and gradient before writing a line equation.
Treat a zero-gradient tangent and its vertical normal separately.
Find every permitted contact point, not just one.

At \(P(a,f(a))\), the tangent is \(y-f(a)=f^{\prime}(a)(x-a)\). For \(m\ne0\), the normal gradient is \(\displaystyle - \frac{1}{m}\). If \(m=0\), the normal is \(x=a\). An external point need not be a point of contact.

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Increasing, decreasing and stationary points

Use signs of the first derivative and give complete coordinates.
Classify maxima and minima with a valid test.
Distinguish local and absolute extrema; include endpoint candidates.

Find stationary candidates from \(f^{\prime}(x)=0\), then use the original function for their ordinates. A + to − derivative sign change gives a maximum; − to + gives a minimum. At a stationary point, \(f^{\prime\prime}>0\) gives a minimum and \(f^{\prime\prime}<0\) a maximum; zero is inconclusive. Compare endpoints for absolute extrema on a closed interval. No classification of points of inflexion is required here.

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Connected rates, signs and small changes

Link rates with the chain rule and keep signs and units.
Recover a length rate from a volume or area rate.
Distinguish an instantaneous rate from an exact finite change.

Use \(\displaystyle \frac{dy}{dt}=\frac{dy}{dx}\frac{dx}{dt}\) at the specified instant. A decreasing quantity has a negative signed rate. Convert units before substitution. In the explicitly tagged small-change applications, \(\Delta y\approx f^{\prime}(x)\Delta x\), not an exact equality.

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Optimisation with feasible domains

Eliminate a variable and state the allowed interval.
Reject infeasible stationary values and justify a global optimum.
Return every requested dimension, value and unit.

Build a one-variable objective and state its feasible domain. Differentiate; test all allowed stationary candidates and relevant endpoints. A local second-derivative test alone does not compare every feasible value. For integer inputs, compare the neighbouring allowed integers around a continuous optimum.

Read the full notes and 3 worked examples · Next: short practice