Before the midterm

Treating the base or the exponent as a constant

The habit: When both parts of a power vary with x, take logarithms first, so that each part contributes its own term.

6 practice questions are built to catch it, in 1 topic.

Practise against it: 5 questions

What it looks like

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

Challenging4 marks · 6 min

Let $f(x) = \bigl(x^{2} + 1\bigr)^{x}$ . Use logarithmic differentiation to find $f'(1)$ .

With the slip: $\displaystyle 2$
Worked through: $\displaystyle f'(1) = 2\log 2 + 2$

x(x² + 1)^x−1 · 2x, the power rule with the exponent frozen, is 2 at x = 1. The exponent x varies too and contributes a second term.

See the worked solution

Where it comes up

The chain rule and logarithms6 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 $f(x) = x^{\cos x}$ for $x > 0$ . Find $f'\!\left(\dfrac{\pi}{3}\right)$ .
The chain rule and logarithmsChallenging
02
Let $f(x) = (2 + \cos x)^{x}$ . Find $f'\!\left(\dfrac{\pi}{2}\right)$ .
The chain rule and logarithmsChallenging
03
Let $g(x) = x^{x} + 3^{x} + x^{3}$ for $x > 0$ . Find $g'(x)$ .
The chain rule and logarithmsChallenging
04
Let $f(x) = x^{\sin x}$ for $x > 0$ . Use logarithmic differentiation to find $f'(x)$ .
The chain rule and logarithmsChallenging
05
Let $f$ be a differentiable function with $f(x) > 0$ for every $x$ , $f(1) = e$ and $f'(1) = 3$ . Let $g$ be the function below. Find $g'(1)$ . $\displaystyle g(x) = \bigl(f(x^{2})\bigr)^{x}$
The chain rule and logarithmsCapstone