Higher Maths: Chain Rule, Tangents and Curve Sketching

Differentiation is one of the largest parts of the calculus skills in Higher Mathematics, and it turns up in both question papers. This material covers the first two groups of calculus skills listed in the Higher Mathematics course specification: differentiating functions, and using differentiation to investigate the nature and properties of functions.

It starts with the techniques. You rewrite a function such as $\frac{x^{2} + 3}{\sqrt{x}}$ as a sum of powers of $x$ before differentiating, differentiate $k \sin x$ and $k \cos x$ with the sign of the cosine derivative handled correctly, and use the chain rule on composite functions: a bracket raised to a power, a reciprocal such as $\frac{1}{(2 x + 1)^{3}}$ , $\cos^{2} x$ and $\cos 3 x$ . Radians are used throughout, and exact values such as $\sin \frac{π}{3} = \frac{\sqrt{3}}{2}$ appear in the non-calculator questions.

It then uses the derivative to investigate curves. You find the equation of a tangent at a given point, and learn to tell it apart from the normal. You decide where a function is strictly increasing or decreasing by solving $f^{'} (x) > 0$ , show that a function is never decreasing by writing its derivative as a multiple of a square, and find stationary points and their nature with a nature table, including points of inflexion where the derivative does not change sign, as in a quartic with a repeated factor. The last step is curve sketching: intersections with the axes, stationary points and their nature, and the behaviour of the curve for large positive and negative values of $x$ .

Optimisation, the greatest and least values of a function on a closed interval and rates of change are in the category's separate material on applying differential calculus.

The material offers three formats. The quiz has 12 questions whose explanations show each step and name the slip behind the typical wrong options, such as missing the chain-rule factor or writing the normal instead of the tangent. The flashcards collect the rules to recall: the chain rule, the derivatives of $\sin x$ and $\cos x$ , the tangent method and the sign patterns in a nature table. The printable written work has 8 longer tasks to set out by hand, including a full curve sketch of $y = x^{3} - 3 x^{2}$ , a proof that a function is never decreasing, and a written explanation of why a stationary point need not be a turning point.

It is aimed at S5 and S6 students preparing for Higher Mathematics.

Differentiate expressions that can be written as powers of x
Differentiate k sin x and k cos x
Differentiate composite functions using the chain rule
Find the equation of a tangent to a curve at a given point
Determine where a function is strictly increasing or decreasing
Find stationary points and determine their nature with a nature table, including points of inflexion
Sketch a curve using its axis intercepts, stationary points and behaviour for large x

Practice material written by Zestly, based on the Qualifications Scotland (formerly SQA) Higher Mathematics course specification, version 3.0 (calculus skills: differentiating functions; using differentiation to investigate the nature and properties of functions).

Sample question

Given $f(x) = \dfrac{x^2 + 3}{\sqrt{x}}$ for $x > 0$, find $f'(x)$.

$\frac{3}{2}x^{\frac{1}{2}} - \frac{3}{2}x^{-\frac{3}{2}}$
$\frac{2x}{\frac{1}{2}x^{-\frac{1}{2}}}$
$\frac{3}{2}x^{\frac{1}{2}} + \frac{3}{2}x^{-\frac{3}{2}}$
$\frac{1}{2}x^{-\frac{1}{2}} - 3x^{-\frac{3}{2}}$

See the answer

$\frac{3}{2}x^{\frac{1}{2}} - \frac{3}{2}x^{-\frac{3}{2}}$

First write the function as powers of $x$: $f(x) = x^{2 - \frac{1}{2}} + 3x^{-\frac{1}{2}} = x^{\frac{3}{2}} + 3x^{-\frac{1}{2}}$. Then $f'(x) = \frac{3}{2}x^{\frac{1}{2}} + 3 \times \left(-\frac{1}{2}\right)x^{-\frac{3}{2}} = \frac{3}{2}x^{\frac{1}{2}} - \frac{3}{2}x^{-\frac{3}{2}}$. Differentiating the top and bottom separately is not a valid rule.

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