Before the midterm

Treating one factor of a product as a constant

The habit: When both pieces change, the derivative has two terms: u′v + uv′.

17 practice questions are built to catch it, in 9 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.

Standard3 marks · 4 min

The point $(1, 2)$ lies on the curve below. Find $\dfrac{dy}{dx}$ at that point.

x^{2} + xy + y^{2} = 7

With the slip: $\displaystyle -1$
Worked through: $\displaystyle \left.\frac{dy}{dx}\right|_{(1,\,2)} = -\frac{4}{5}$

That differentiates xy as y alone, giving y′ = −(2x + y)/(2y) = −1. The x dy/dx from xy puts x + 2y = 5 underneath, not 2y = 4.

See the worked solution

Where it comes up

Differentiation rules2 questions
Exponential and trigonometric functions1 question
Implicit differentiation and inverse trigonometric functions5 questions
Related rates2 questions
Optimization1 question
Linear approximation1 question
Taylor polynomials1 question
Newton’s method1 question
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
A student finds $f'(1)$ for the function below as shown. Find the first line that is wrong. $\displaystyle f(x) = (x^{2} + 1)(x^{3} - 2x)$
Differentiation rulesStandard
02
Let $f(x) = e^{x}(\sin x - \cos x)$ . Find $f'(x)$ .
Exponential and trigonometric functionsStandard
03
Find the $x$ -intercept of the line tangent to the curve $y = x\arctan x$ at the point where $x = 1$ .
Implicit differentiation and inverse trigonometric functionsStandard
04
The point $(2, 2)$ lies on the curve below. Find an equation of the normal line to the curve there: the line through $(2, 2)$ perpendicular to the tangent line. $\displaystyle y^{3} + xy = 12$
Implicit differentiation and inverse trigonometric functionsStandard
05
Find $\dfrac{dy}{dx}$ in terms of $x$ and $y$ for the curve below. $\displaystyle xy^{2} + e^{y} = 2x$
Implicit differentiation and inverse trigonometric functionsStandard
06
The table gives values at $x = 2$ of two twice-differentiable functions $f$ and $g$ and of their first and second derivatives. Let $h(x) = f(x)\,g(x)$ . Find $h''(2)$ .
Differentiation rulesChallenging