Differentiate

Choose the rule

In 16 of 20 recent exams
3% of the marks
Paper 1 or 2
2 to 5 marks

How it is set in the exam

Paper 1 (no calculator), 2–5 marks, and part (a) of many Paper 2 (calculator) questions. The most common calculus part of all: a stand-alone "find $f'(x)$ " or "show that $f'(x) = \ldots$ " is in most papers, and differentiating is the first step of almost every other calculus question.

Spot it in the exam

Exam phrases to spot (6)

“find $f'(x)$ ”
“differentiate”
“find $\frac{dy}{dx}$ ”
“find the derivative of …”
“show that $f'(x) = \ldots$ ”
“show that $f''(x) = \ldots$ ”

In disguise (4)

"the rate of change of the gradient": that is $f''$ , so differentiate twice
$e^{2\ln x}$ or $\ln\left(e^{3x}\right)$ : simplify first (to $x^2$ and $3x$ ), and the rest is one line
values instead of formulas: in a table, in the text, or read off a graph. For $h(x) = x f(x^2)$ with $f(4) = 3$ and $f'(4) = -1$ : $h'(x) = f(x^2) + 2x^2 f'(x^2)$ , so $h'(2) = 3 + 8(-1) = -5$
a transformation in words: "the graph of $g$ is the graph of $f$ stretched vertically by a factor of $3$ , then translated by $\binom{-2}{-1}$ " means $g(x) = 3f(x + 2) - 1$ , so $g'(x) = 3f'(x + 2)$ (the formula from the words: Transform a graph)

Looks like it, but isn't (3)

the average rate of change between two points: a chord gradient $\frac{f(b) - f(a)}{b - a}$ , with no derivative
a rate is given and a total is asked: integrate (Total change from a rate)
the composite itself is asked for, $f(g(x))$ as an expression, with no derivative: Form a composite function and use it

Worked example

Split the fraction, then powers

Paper 1 (no calculator)

Question

Let $f(x) = \dfrac{2x^3 - 3\sqrt{x} + 4}{x}$ , for $x > 0$ . Find $f'(x)$ . [4]

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Which method?

The function is…	Use
a sum of powers, roots and $\frac{1}{x^n}$ , or a fraction over a single term	rewrite as powers (split the fraction), then the power rule
a function of a function: $(3x - 1)^5$ , $e^{x^2}$ , $\sin 3x$ , $\sin^2 x$ , $\ln(2x + 1)$ , $\sqrt{4 - x^2}$	chain rule
a product of two functions of $x$	product rule: write $u$ , $v$ , $u'$ , $v'$
a quotient with a sum (or a non-power) underneath	quotient rule; if the top is a number, a power and the chain rule are quicker
a log of a product, quotient or power; $e$ to a log; a log of $e$	simplify first (log laws; $e^{2\ln x} = x^2$ ), then term by term
values only (a table, the text, a graph, a transformation)	write the rule in function notation, then read off the values

Steps

Read the structure:
the last operation is the outermost rule (how to choose).
Rewrite if it helps:
powers, log laws, $e^{\ln}$ .
Apply the rule.
For products and quotients, write $u$ , $v$ , $u'$ , $v'$ first. For a chain: the outside's derivative, the inside unchanged, times the inside's derivative. For $f''$ , differentiate $f'$ again, starting from its simplest form.
Simplify only as far as asked.
"Find" accepts any correct form. "Show that" ends exactly on the given form: work forwards from $f$ and show the key step (the common factor, the cancelling). If $f'(x) = 0$ or the sign of $f'$ comes next, factorise.
Values only:
write the rule in function notation first, e.g. $h'(1) = f'(g(1))\,g'(1)$ , then substitute.

Common mistakes

Forgetting the inner derivative
$\frac{d}{dx}e^{3x} = 3e^{3x}$ , not $e^{3x}$ ; $\;\frac{d}{dx}e^{x^2} = 2xe^{x^2}$ .
Using the power rule on an exponential
$e^{3x}$ is not a power of $x$ , so its derivative is not $3x\,e^{3x - 1}$ .
The sign of $\cos$
$\frac{d}{dx}\cos x = -\sin x$ .

More mistakes (7)

Quotient rule order
the top is $v u' - u v'$ . The other order gives the negative of the answer.
Skipping the product rule
$\frac{d}{dx}\left(x^2e^x\right) = 2xe^x + x^2e^x$ , not $2xe^x$ .
Rewriting fractions
$\frac{1}{3x^2} = \frac{1}{3}x^{-2}$ , not $3x^{-2}$ .
ln of a multiple
$\frac{d}{dx}\ln 3x = \frac{1}{x}$ , not $\frac{3}{x}$ . And $\ln(x + 1)$ is not $\ln x + \ln 1$ .
Values only: reading $f'$ at $x$ instead of at $g(x)$ .
$h'(1) = f'(g(1))\,g'(1) = f'(3)\,g'(1)$ .
Stopping a "show that" one line early
end on exactly the given form. In Example 1.3 the $x$ has to be seen cancelling.
Degree mode
for calculus with trig: the answer scores ().

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