- Difficulty
- Introductory
- Suggested time
- 3 min
- Marks
- 2
The chain rule and logarithms practice question 7
The table lists values of two differentiable functions, and , and of their derivatives. Let . Find .
In , which function acts first? Decide at which input each derivative must be read before you go to the table.
By the chain rule, . Read from the table first: that is where is needed.
An exact value.
Theory UBC Math 100 §4.3CLP-1 §2.9Key results
Worked solution
Strategy
You never see a formula for f or g; the chain rule needs only the inner value g(1), the outer derivative there, and the inner derivative at 1.
Working
Chain rule: the outer derivative at the inner value, times the inner derivative.
Read the inner function and its derivative at x = 1.
f′ is read at 4, the value of g(1), not at 1.
Answer
The answer is negative 6.
Check
As x passes 1, g rises through g(1) = 4, where f is falling, so h must be decreasing at x = 1 and h′(1) must be negative.
Where this goes wrong
Reading f′ at 1 instead of at g(1) = 4, which gives 5 · 3 = 15. The outer derivative is evaluated where the inner function lands.
Leans on
Composition of functions, from Functions, graphs and inverses
Try a similar question
Let , where is a differentiable function with and . Find .Saved in this browser only.