- Difficulty
- Standard
- Suggested time
- 4 min
- Marks
- 3
The chain rule and logarithms practice question 13
Let . Find .
A quotient whose numerator is an exponential of , not of . Which rule handles the fraction, and what extra factor does that exponent bring?
Quotient rule, with the chain rule inside it: the derivative of is .
At the denominator is 2 and its derivative is 2. The quotient rule divides by the square of the denominator.
An exact expression, or a decimal to two decimal places.
Theory UBC Math 100 §4.3CLP-1 §2.9Key results
Worked solution
Strategy
Quotient rule outside, chain rule on the exponential, then substitute x = 1.
Working
u′v − uv′ over v², with u′ = 2e2x from the chain rule and v′ = 2x.
At x = 1: 1 + x² = 2 and 2x = 2.
Answer
The answer is e squared over 2, about 3.69.
Check
At x = 1, log f = 2x − log(1 + x²) has slope 2 − 1 = 1, so f′(1) must equal f(1) = e²/2.
Where this goes wrong
Differentiating e2x as e2x and losing the factor 2. At x = 1 the two terms of the numerator then cancel and the answer comes out as 0. The exponent 2x has derivative 2, and the chain rule multiplies by it.
Leans on
The quotient rule, from Differentiation rules
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