Question
Find the derivative of the function. \[ f(x)=\arcsin (x-2) \] \( f^{\prime}(x)=\square \)
Ask by Colon Davison. in the United States
Feb 26,2025
Upstudy AI Solution
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Answer
The derivative of \( f(x) = \arcsin(x - 2) \) is \( f'(x) = \frac{1}{\sqrt{1 - (x - 2)^2}} \).
Solution
To differentiate f(x) = arcsin(x − 2), we use the chain rule. Recall that the derivative of arcsin(u) with respect to u is
d/du (arcsin(u)) = 1/√(1 − u²).
For our function, u = x − 2, and its derivative with respect to x is 1. Therefore, applying the chain rule gives
f '(x) = (1/√(1 − (x − 2)²)) · 1 = 1/√(1 − (x − 2)²).
Thus, the derivative is
f '(x) = 1/√(1 − (x − 2)²).
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Simplify this solution Bonus Knowledge
To find the derivative of the function \( f(x) = \arcsin(x - 2) \), we can use the chain rule. The derivative of \( \arcsin(u) \) is \( \frac{1}{\sqrt{1 - u^2}} \), where \( u = x - 2 \). Thus, we have: 1. First, we find \( u' = 1 \) since the derivative of \( u = x - 2 \) with respect to \( x \) is simply 1. 2. Therefore, the derivative \( f'(x) \) is given by: \[ f'(x) = \frac{1}{\sqrt{1 - (x - 2)^2}} \cdot 1 = \frac{1}{\sqrt{1 - (x - 2)^2}} \] So, \[ f'(x) = \frac{1}{\sqrt{1 - (x - 2)^2}} \] is the final answer! Be cautious with the domain of arcsin to ensure \( -1 \leq x - 2 \leq 1 \).
