Before the midterm

Treating the factors separately

The habit: A product or quotient is not differentiated, or antidifferentiated, factor by factor: use the rule, or spot the chain rule in reverse.

18 practice questions are built to catch it, in 6 topics.

See it fail in a figure: The slip is the corner, in the differentiation rules lesson

Practise against it: 6 questions

What it looks like

A question from differentiation rules, with the answer this slip gives beside the right one.

Introductory2 marks · 3 min

Let $f(x) = x^{2}\sqrt{x} - \dfrac{6}{\sqrt[3]{x}}$ for $x > 0$ . Find $f'(x)$ .

With the slip: $\displaystyle \sqrt{x} + 2 x^{\frac{-4}{3}}$
Worked through: $\displaystyle f'(x) = \frac{5}{2}x^{3/2} + 2x^{-4/3}$

√x is (x²)′ · (√x)′ = 2x · 1/(2√x). Combine first: x²√x = x^5/2, and differentiate that power.

See the worked solution

Where it comes up

The definition of the derivative1 question
Differentiation rules8 questions
Exponential and trigonometric functions4 questions
The chain rule and logarithms3 questions
Curve sketching1 question
Functions of two variables and partial derivatives1 question

Practise against it

Each of these has an answer this slip gives. Work them on paper: if you fall into it, the verdict names it and the question comes back in review.

01
Let $f(x) = \dfrac{x^{3} - 2x + 5}{x^{2}}$ . Find $f'(x)$ .
Differentiation rulesIntroductory
02
Let $f(x) = x\,|x|$ . Which statement about $f$ at $x = 0$ is correct?
The definition of the derivativeStandard
03
Let $h(x) = (2x + 1)(x^{2} - 3)$ . Find $h'(2)$ .
Differentiation rulesStandard
04
Let $q(x) = \dfrac{x^{2} + 1}{x - 2}$ . Which expression is $q'(x)$ ?
Differentiation rulesStandard
05
The table gives values of two differentiable functions $f$ and $g$ and of their derivatives. Let $Q(x) = \dfrac{f(x)}{g(x)}$ . Find $Q'(3)$ .
Differentiation rulesStandard
06
Let $f(x) = \dfrac{3x^{2} + 5}{2 - x}$ . Find $f'(-1)$ .
Differentiation rulesStandard