Before the midterm

Misremembering a standard formula

The habit: Check the table: log x differentiates to 1/x, a^x to a^x log a, and 1/x antidifferentiates to log|x|.

28 practice questions are built to catch it, in 11 topics.

Practise against it: 6 questions

What it looks like

A question from implicit differentiation and inverse trigonometric functions, with the answer this slip gives beside the right one.

Introductory2 marks · 3 min

Let $f(x) = \arccos\!\left(\dfrac{x}{2}\right)$ . Find $f'(1)$ .

With the slip: $\displaystyle -0.4$
Worked through: $\displaystyle f'(1) = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3}$

−0.4 is −(1/2)/(1 + 1/4), built on 1 + u², the shape of the arctan derivative. For arccos it is −u′/√(1 − u²).

See the worked solution

Where it comes up

Functions, graphs and inverses1 question
The definition of the derivative1 question
Differentiation rules2 questions
Exponential and trigonometric functions6 questions
The chain rule and logarithms4 questions
Implicit differentiation and inverse trigonometric functions8 questions
Related rates2 questions
Linear approximation1 question
Optimization in two variables and Lagrange multipliers1 question
The Mean Value Theorem1 question
Antiderivatives1 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
Find the exact value of the expression below. $\displaystyle \tan\!\left(\frac{5\pi}{6}\right)\sec\!\left(\frac{\pi}{3}\right)$
Functions, graphs and inversesIntroductory
02
The points $P(1, 1)$ , $Q(4, 3)$ and $R(k, -3)$ are the vertices of a triangle with a right angle at $Q$ . Find $k$ .
The definition of the derivativeIntroductory
03
Let $f(x) = 3x^{4} - 5x^{2} + 7$ . Find $f'(x)$ .
Differentiation rulesIntroductory
04
Which of the following is the derivative of $g(x) = 3^{x}$ ?
Exponential and trigonometric functionsIntroductory
05
Let $f(x) = 3\tan x - 2\sec x$ . Find $f'\!\left(\dfrac{\pi}{6}\right)$ .
Exponential and trigonometric functionsIntroductory
06
The normal line to the curve $y = 2x^{2} - 3x$ at the point $(2, 2)$ is the line through $(2, 2)$ perpendicular to the tangent line there. It meets the curve at a second point. Find the $x$ -coordinate of that point.
Differentiation rulesChallenging