- Difficulty
- Standard
- Suggested time
- 4 min
- Marks
- 2
Implicit differentiation and inverse trigonometric functions practice question 30
A student finds for the function below as shown. Find the first line that is wrong.
Redo each line yourself: the derivative from the standard formula and the chain rule, then the simplification, then the value at .
The derivatives of arcsin and arctan both involve u²: one has 1 − u² under a square root, the other has 1 + u² with no root. Which is which?
Differentiate implicitly and see what comes out as.
Theory UBC Math 100 §4.7CLP-1 §2.12Key results
Corrected working
Strategy
Use the derivative of arctan with the chain rule, or differentiate tan y = 2x implicitly.
Working
The derivative of arctan u is 1/(1 + u²), times u′.
Substitute x = 1.
Answer
The answer is y prime of 1 equals two fifths.
Check
y(0.99) = arctan 1.98 ≈ 1.1031 and y(1.01) = arctan 2.02 ≈ 1.1111, a rise of 0.008 over a step of 0.02, so a slope of 0.4, not 2/√5 ≈ 0.894.
Where this goes wrong
Putting a square root under the 1 + u². That root belongs to the derivative of arcsin; the derivative of arctan has none.
Leans on
The chain rule, from The chain rule and logarithms
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