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Longhand

Before the midterm

Evaluating the outer derivative at the wrong value

The habit: The outside function’s derivative is taken at the output of the inside, not at x.

4 practice questions are built to catch it, in 2 topics.

See it fail in a figure: Read the outside at Q, in the chain rule and logarithms lesson

Practise against it: 3 questions

What it looks like

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

Introductory2 marks · 3 min

The table lists values of two differentiable functions, ff and gg, and of their derivatives. Let h(x)=f(g(x))h(x) = f\bigl(g(x)\bigr). Find h′(1)h'(1).

x124f(x)279f′(x)51−2g(x)438g′(x)37−1\begin{array}{c|ccc} x & 1 & 2 & 4 \\ \hline f(x) & 2 & 7 & 9 \\ f'(x) & 5 & 1 & -2 \\ g(x) & 4 & 3 & 8 \\ g'(x) & 3 & 7 & -1 \end{array}
With the slip
15\displaystyle 15
Worked through
h′(1)=−6\displaystyle h'(1) = -6

15 is f′(1) · g′(1) = 5 · 3, with f′ read at 1. f′ belongs at the inner value, g(1) = 4: read f′ there instead.

See the worked solution

Where it comes up

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.

  1. 01
    A student finds g′(1)g'(1) for the function below as shown. Find the first line that is wrong. g(x)=(2x+1)5\displaystyle g(x) = (2x + 1)^{5}

    The chain rule and logarithmsStandard

  2. 02
    The revenue from selling qq units of a product is R(q)=60q−0.5q2R(q) = 60q - 0.5q^{2} dollars. Sales are growing: tt weeks from now the company expects to be selling q(t)=20+4tq(t) = 20 + 4t units. How fast is the revenue changing 5 weeks from now?

    The chain rule and logarithmsStandard

  3. 03
    Let f(x)=arcsin⁡xf(x) = \arcsin\sqrt{x} for 0<x<10 < x < 1. Find f′(x)f'(x).

    Implicit differentiation and inverse trigonometric functionsStandard

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