- Difficulty
- Introductory
- Suggested time
- 2 min
- Marks
- 2
Implicit differentiation and inverse trigonometric functions practice question 7
Let . Find .
Which function is outside and which is inside in ? Recall the derivative of arctan, then deal with the inside.
. Here .
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 §4.7CLP-1 §2.12Key results
Worked solution
Strategy
The derivative of arctan, taken at the inside 3x, times the derivative of the inside.
Working
Outer derivative 1/(1 + u²) at u = 3x, then the inner derivative.
The inner derivative is 3, and (3x)² = 9x².
Answer
The answer is f prime of x equals 3 over (1 plus 9 x squared).
Check
Near x = 0, arctan(3x) ≈ 3x, so f′(0) should be 3, and f′ should fade to 0 as arctan levels off; 3/(1 + 9x²) does both.
Where this goes wrong
Leaving out the inner derivative, which gives 1/(1 + 9x²). The inside is 3x, so the chain rule multiplies by 3.
Leans on
The chain rule, from The chain rule and logarithms
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