Before the midterm

Nesting the functions the wrong way

The habit: Work from the inside out: say which function is applied first, then substitute back to check.

7 practice questions are built to catch it, in 4 topics.

Practise against it: 6 questions

What it looks like

A question from the chain rule and logarithms, with the answer this slip gives beside the right one.

Standard3 marks · 4 min

Let $f(x) = \sin^{3}\!\left(\pi x^{2}\right)$ . Find $f'(x)$ .

With the slip: $\displaystyle 6 \pi x \cos^{2}\left(\pi x^{2}\right)$
Worked through: $\displaystyle f'(x) = 6\pi x\,\sin^{2}\!\left(\pi x^{2}\right)\cos\!\left(\pi x^{2}\right)$

That is 3 cos²(πx²) · 2πx: the sine was swapped for its derivative inside the cube. The cube’s derivative keeps the sine as it is, 3 sin²(πx²), and the cosine comes in as a separate factor.

See the worked solution

Where it comes up

Functions, graphs and inverses2 questions
The chain rule and logarithms3 questions
Implicit differentiation and inverse trigonometric functions1 question
Taylor polynomials1 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 $f(x) = 2x - 5$ and $g(x) = x^{2} + 1$ . Evaluate the composition below. $\displaystyle (f \circ g)(3)$
Functions, graphs and inversesIntroductory
02
The table lists values of two differentiable functions, $f$ and $g$ , and of their derivatives. Let $h(x) = f\bigl(g(x)\bigr)$ . Find $h'(1)$ .
The chain rule and logarithmsIntroductory
03
Let $h(x) = \sqrt{1 + e^{2x}}$ . Suppose $h(x) = f\bigl(g(x)\bigr)$ with the inner function $g(x) = e^{2x}$ . Which formula gives the outer function $f$ ?
Functions, graphs and inversesStandard
04
Let $f(x) = \cos^{2}\!\left(x^{3}\right)$ . Find $f'(x)$ .
The chain rule and logarithmsStandard
05
Let $g(x) = \arctan\!\left(\dfrac{1}{x}\right)$ for $x \neq 0$ . Each graph below is drawn with dashed lines at $y = \dfrac{\pi}{2}$ and $y = -\dfrac{\pi}{2}$ . Which one is the graph of $g$ ?
Implicit differentiation and inverse trigonometric functionsChallenging
06
Let $f(x) = x^{2}\cos\bigl(x^{2}\bigr)$ . Find $f^{(10)}(0)$ , the tenth derivative of $f$ at $0$ .
Taylor polynomialsChallenging

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