- Difficulty
- Standard
- Suggested time
- 4 min
- Marks
- 2
The chain rule and logarithms practice question 30
A student finds for the function below as shown. Find the first line that is wrong.
Check every line against what it claims: the derivative against the chain rule, each substitution against the expression it starts from, and the arithmetic.
In line 1 the outer derivative is with . What is when ?
Substitute x = 1 into the whole of line 1, not into one piece of it.
Theory UBC Math 100 §4.3CLP-1 §2.9Key results
Corrected working
Strategy
Chain rule, then substitute x = 1 into the whole derivative.
Working
Outer derivative at the inside, times the inner derivative.
2(1) + 1 = 3.
Answer
The answer is g prime of 1 equals 810.
Check
g(0.999) ≈ 242.191 and g(1.001) ≈ 243.811, a rise of about 1.62 over a step of 0.002, so a slope of 810, nowhere near 10.
Where this goes wrong
Evaluating the outer derivative at x instead of at the inner value. The 5u⁴ in the chain rule wants u = 2x + 1, which is 3 at x = 1.
Leans on
The power rule, from Differentiation rules
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