- Difficulty
- Introductory
- Suggested time
- 3 min
- Marks
- 2
The chain rule and logarithms practice question 6
Let for . Find .
Read carefully: is the cube applied to , or to the value of ?
means : a cube on the outside, inside. Power rule first, then multiply by the derivative of .
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 §4.3CLP-1 §2.9Key results
Worked solution
Strategy
A power of a trigonometric function is a composition: the cube is applied to the value of tan x. The chain rule gives the power rule on the outside times the derivative of tan x.
Working
The power sits on the value of tan x, not on x.
Power rule on the outside, with tan x left as it is.
The derivative of tan x is sec² x.
Answer
The answer is 3 tan squared x times sec squared x.
Check
At π/4 the formula gives 6, and tan³x climbs from about 0.942 to 1.062 between π/4 − 0.01 and π/4 + 0.01, a slope of about 6.
Where this goes wrong
Stopping at 3 tan² x. The cube is applied to tan x, so the chain rule also multiplies by the derivative of tan x, which is sec² x.
Leans on
Derivatives of the trigonometric functions, from Exponential and trigonometric functions
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