Rules of Differentiation

Q: What notation should I use for derivatives?

Match the question: $f'(x)$ for a function named $f$, $\frac{dy}{dx}$ for $y = \ldots$, and $D_x[\ldots]$ when the expression has no name.

The rules of differentiation let you find a derivative in one line instead of a page of first principles. Almost every calculus question in Paper 1 depends on them, and most errors come from not rewriting the expression properly before you start.

The power rule

If $f (x) = x^{n}$ for any real number $n$ , then

$f^{'} (x) = n x^{n - 1}$

Multiply by the power, then reduce the power by 1.

Examples.

$\frac{d}{d x} (x^{5}) = 5 x^{4}$
$\frac{d}{d x} (x^{- 2}) = - 2 x^{- 3}$
$\frac{d}{d x} (x^{\frac{1}{2}}) = \frac{1}{2} x^{- \frac{1}{2}}$

The derivative of a constant is 0, because a horizontal line has gradient 0.

Sum and constant rules

Constant multiple rule: $\frac{d}{d x} [k f (x)] = k f^{'} (x)$

Sum and difference rule: $\frac{d}{d x} [f (x) \pm g (x)] = f^{'} (x) \pm g^{'} (x)$

So you differentiate a polynomial term by term.

Example. $y = 4 x^{3} - 6 x^{2} + 7 x - 9$

$\frac{d y}{d x} = 12 x^{2} - 12 x + 7$

There is no product or quotient rule in the CAPS syllabus. A product or quotient must be expanded or simplified first.

Rewriting before differentiating

Before you differentiate, every term must be in the form $a x^{n}$ . That means:

Roots become fractional exponents: $x = x^{\frac{1}{2}}$ and $3 x^{2} = x^{\frac{2}{3}}$
Fractions with $x$ in the denominator become negative exponents: $\frac{4}{x ^{3}} = 4 x^{- 3}$
Products are expanded: $(2 x - 1)^{2} = 4 x^{2} - 4 x + 1$
Quotients are split: $\frac{x ^{3} + 2 x}{x} = x^{2} + 2$ , for $x \neq = 0$

Example. $f (x) = \frac{x ^{2} - 3 x}{x}$

$f (x) = x - 3 x^{- \frac{1}{2}}$

$f^{'} (x) = 1 + \frac{3}{2} x^{- \frac{3}{2}}$

Notation matters. $f^{'} (x)$ , $\frac{d y}{d x}$ and $D_{x} [\dots]$ are accepted. Writing $\frac{d y}{d x}$ when the function is named $f$ can lose a mark. Leave final answers with positive exponents unless the question says otherwise.

Key terms

Power rule: $\frac{d}{d x} (x^{n}) = n x^{n - 1}$ for any real $n$ .
$D_{x}$: Notation meaning 'differentiate with respect to $x$ '.
$\frac{d y}{d x}$: The derivative of $y$ with respect to $x$ , used when the function is written as $y = \dots$
Term: A part of an expression separated by $+$ or $-$ signs. Polynomials are differentiated term by term.

Common mistakes

Differentiating a product such as $(x + 2) (x - 3)$ by multiplying the derivatives. Expand first.
Writing $\frac{1}{x ^{2}}$ as $x^{2}$ or $x^{- \frac{1}{2}}$ . It is $x^{- 2}$ .
Splitting a fraction only over part of the numerator, for example $\frac{x ^{2} + 4}{x} = x + 4$ .
Forgetting that the derivative of a constant is 0.
Mixing notation, such as writing $f^{'} (x)$ when the question uses $y$ .

Practice questions

Reveal steps one at a time, then tick the marks you earned.

Question 1

3 marks total

knowledge

Determine $\frac{d y}{d x}$ if $y = 4 x^{3} - 7 x + 2$ .

Worked solution

Hidden - try the question first

Solution

$\frac{d y}{d x} = 12 x^{2}$
$- 7$
$+ 0$ , so $\frac{d y}{d x} = 12 x^{2} - 7$

Question 2

5 marks total

routine

Determine $\frac{d y}{d x}$ if $y = 6 x - \frac{2}{x ^{3}}$ . Leave your answer with positive exponents.

Worked solution

Hidden - try the question first

Solution

$\frac{d y}{d x} = 12 x^{2}$
$\frac{d y}{d x} = 3 x^{- \frac{1}{2}} + 6 x^{- 4}$
$= \frac{3}{x} + \frac{6}{x ^{4}}$

Question 3

6 marks total

complex

Given $f (x) = \frac{2 x ^{3} - 5 x ^{2} + 4}{x ^{2}}$ , $x \neq = 0$ .

Determine $f^{'} (x)$ and hence calculate $f^{'} (2)$ .

Worked solution

Hidden - try the question first

Solution

$\frac{d y}{d x} = 12 x^{2}$
$- 7$
$= 2 - \frac{8}{x ^{3}}$
$f^{'} (2) = 2 - \frac{8}{8} = 1$

Question 4

8 marks total

problem-solving

Given $f (x) = a x^{3} + b x^{2}$ , where $f^{'} (1) = 0$ and $f^{'} (2) = 12$ .

(a) Determine the values of $a$ and $b$ . (5)

(b) Hence determine the values of $x$ for which $f^{'} (x) = 36$ . (3)

Worked solution

Hidden - try the question first

Solution

(a) $f^{'} (x) = 3 a x^{2} + 2 b x$
$f^{'} (1) = 3 a + 2 b = 0$ and $= 2 - \frac{8}{x ^{3}}$
From the first equation $f^{'} (2) = 2 - \frac{8}{8} = 1$ , so $12 a - 6 a = 12$ and $a = 2$
$b = - 3$
(b) $f^{'} (x) = 6 x^{2} - 6 x = 36$
$x^{2} - x - 6 = 0 \Rightarrow (x - 3) (x + 2) = 0$
$x = 3$ or $x = - 2$

FAQs

Is the product rule in the IEB Grade 12 syllabus?

No. CAPS has no product, quotient or chain rule. Expand products and split quotients before differentiating.

Do I have to give my derivative with positive exponents?

It is good practice and some memos expect it. If a question says 'leave your answer with positive exponents', it is required.

What notation should I use for derivatives?

Match the question: $f^{'} (x)$ for a function named $f$ , $\frac{d y}{d x}$ for $y = \dots$ , and $D_{x} [\dots]$ when the expression has no name.

The power rule

Sum and constant rules

Rewriting before differentiating

Key terms

Common mistakes

Practice questions

FAQs

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