- Difficulty
- Standard
- Suggested time
- 4 min
- Marks
- 3
The chain rule and logarithms practice question 19
Let . Find .
How many layers does have, and in what order are they applied?
means : a square, then a cosine, then .
One factor per layer, each taken at what is inside it: from the square, then one from the cosine, then one from .
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
Three layers, so the chain rule is used twice. Differentiate from the outside in, leaving everything inside each layer untouched.
Working
The square is outermost, then the cosine, then x³.
One factor per layer, each evaluated at what is inside it.
Collect the constants: 2 · (−1) · 3 = −6. It can also be written −3x² sin(2x³).
Answer
The answer is negative 6 x squared times cosine of x cubed times sine of x cubed.
Check
Near x = 0, cos²(x³) ≈ 1 − x⁶, whose derivative is −6x⁵, and the formula is ≈ −6x² · x³ = −6x⁵ there too.
Where this goes wrong
Writing −2 cos(x³) sin(x³) and stopping, which leaves out the derivative of x³. Three layers need three factors.
Leans on
Derivatives of the trigonometric functions, from Exponential and trigonometric functions
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