- Difficulty
- Standard
- Suggested time
- 5 min
- Marks
- 3
The chain rule and logarithms practice question 16
Let . Find .
This is a product, and the second factor is itself a composition. You need the product rule and the chain rule together.
At the inner argument is . Recall and .
An exact expression, or a decimal to two decimal places.
Theory UBC Math 100 §4.3CLP-1 §2.9Key results
Worked solution
Strategy
Apply the product rule, using the chain rule on the trigonometric factor. Evaluate the angle first so the trigonometric values are exact.
Working
The derivative of sin(3x) picks up a factor of 3 from the chain rule.
Combine with u′v + uv′.
Evaluate the angle before the trigonometric functions. The sine term vanishes.
Only the second term survives, and it carries the chain-rule factor 3.
Answer
The answer is negative π²/3, approximately −3.29.
Check
g(π/3) = 0, and g falls from about 0.0323 at π/3 − 0.01 to −0.0335 at π/3 + 0.01, a slope of about −3.29.
Where this goes wrong
Differentiating sin(3x) as cos(3x) and losing the factor of 3, which gives −π²/9. The chain rule multiplies by the derivative of the inside, and at this angle that factor sits in the only term that survives.
Leans on
Derivatives of sine and cosine, from Exponential and trigonometric functions
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