- Topic
- Antiderivatives
- Difficulty
- Standard
- Suggested time
- 4 min
- Marks
- 3
- Theory
- CLP-1 §4.1Key results
Antiderivatives practice question 5
A function satisfies and . Find .
Which standard derivative has the shape one over one plus a square? Look at what is being squared here.
is the derivative of . Here , so try .
Differentiate and compare with .
An exact expression, or a decimal to two decimal places.
Theory CLP-1 §4.1Key results
Worked solution
Strategy
Recognise the derivative of arctan with an inner function 3x; the chain rule’s factor of 3 is exactly the 3 in the numerator.
Working
So F(x) = arctan(3x) + C.
Initial condition.
Evaluate.
Answer
The answer is pi over 4, about 0.785.
Check
F′ falls from 3 at x = 0 to 1.5 at x = 1/3, so F(1/3) lies between 1.5 · 1/3 = 0.5 and 3 · 1/3 = 1, and π/4 ≈ 0.785 does.
Where this goes wrong
Writing F(x) = 3 arctan(3x), keeping the 3 in front as well as getting one from the chain rule. Differentiating that gives 9/(1 + 9x²), three times too much.
Leans on
The derivative of arctan, from Implicit differentiation and inverse trigonometric functions
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