- Difficulty
- Standard
- Suggested time
- 3 min
- Marks
- 2
The chain rule and logarithms practice question 15
Let for . Find , simplified.
The logarithm here is taken of , not of . Which rule does a function of a function call for?
Use with .
Now differentiate , keeping track of its sign, and simplify the quotient you get.
A formula in x. Use ^ for powers and brackets for every input: sin(2x), e^(x^2), log(x+1). Give exact values: 1/6, not 0.167.
Theory UBC Math 100 §4.4CLP-1 §2.10Key results
Worked solution
Strategy
A logarithm of a function: the derivative is the inner derivative over the inner function. Then simplify sin over cos.
Working
Outer function log u, whose derivative is 1/u, at u = cos x; then the inner derivative.
The derivative of cos x is −sin x, and sin x over cos x is tan x.
Answer
The answer is negative tan x.
Check
log(cos x) has its maximum 0 at x = 0 and falls away on both sides, matching −tan x: zero at 0, negative to the right, positive to the left.
Where this goes wrong
Writing 1/cos x and stopping, which differentiates the logarithm but not the cosine inside it. The minus sign from the derivative of cos x is the other slip.
Leans on
Derivatives of the trigonometric functions, from Exponential and trigonometric functions
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