- Difficulty
- Introductory
- Suggested time
- 3 min
- Marks
- 2
Implicit differentiation and inverse trigonometric functions practice question 4
Let . Find .
Is arccos increasing or decreasing? That settles the sign of its derivative. Then ask what the inside, , contributes.
. Here , so .
An exact expression, or a decimal to three decimal places.
Theory UBC Math 100 §4.7CLP-1 §2.12Key results
Worked solution
Strategy
The derivative of arccos is the derivative of arcsin with a minus sign; the argument x/2 brings in a chain-rule factor of 1/2.
Working
Derivative of arccos at the inside, times the inner derivative.
At x = 1 the inside is 1/2, and √(3/4) = √3/2.
Answer
The answer is negative one over root 3, about −0.577.
Check
The inverse, x = 2 cos y, has slope −2 sin(π/3) = −√3 at y = π/3, and 1/(−√3) ≈ −0.577 is negative, as it must be for the falling arccos.
Where this goes wrong
Dropping the minus sign, which gives the derivative of arcsin instead. arccos is decreasing on its whole domain, so its derivative, −1/√(1 − u²), is negative, and with u′ = 1/2 > 0 here, so is f′.
Leans on
The chain rule, from The chain rule and logarithms
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