Before the midterm

Dropping the inner derivative

The habit: After differentiating the outside, multiply by the derivative of the inside, every time.

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

See it fail in a figure: The slip: no squeeze, in the chain rule and logarithms lesson

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.

Introductory1 mark · 2 min

Let $f(x) = (2x + 1)^{4}$ . Find $f'(0)$ .

With the slip: $\displaystyle 4$
Worked through: $\displaystyle f'(0) = 8$

4(2x + 1)³ at x = 0 is 4, the derivative of the outside alone. The inside, 2x + 1, contributes its own derivative as a factor.

See the worked solution

Where it comes up

The definition of the derivative1 question
The chain rule and logarithms21 questions
Implicit differentiation and inverse trigonometric functions5 questions
Related rates5 questions
L’Hôpital’s rule4 questions
Curve sketching3 questions
Linear approximation2 questions
Taylor polynomials2 questions
Newton’s method1 question
Slope fields and Euler’s method2 questions
Functions of two variables and partial derivatives3 questions

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 $F(x) = \sin\!\left(x^{2}\right)$ . Which of the following is $F'(x)$ ?
The chain rule and logarithmsIntroductory
02
Let $g(x) = \log\!\left(x^{2}\right)$ for $x > 0$ . Find $g'(3)$ .
The chain rule and logarithmsIntroductory
03
Let $f(x) = \log_{2}\!\left(x^{3} + 1\right)$ . Find $f'(1)$ .
The chain rule and logarithmsIntroductory
04
Let $f(x) = \sqrt{3x^{2} + 4}$ . Find $f'(x)$ .
The chain rule and logarithmsIntroductory
05
Let $f(x) = \tan^{3}x$ for $-\tfrac{\pi}{2} < x < \tfrac{\pi}{2}$ . Find $f'(x)$ .
The chain rule and logarithmsIntroductory
06
Let $f(x) = \sqrt{10 - 3x}$ . Use the definition of the derivative to find $f'(2)$ , and then find an equation of the tangent line to $y = f(x)$ at $x = 2$ .
The definition of the derivativeStandard