Find the Derivative - d/dx arcsin(x-1)

Problem

d(arcsin(x−1))/d(x)

Solution

1. Identify the outer function as arcsin(u) and the inner function as u=x−1

2. Apply the chain rule, which states that d(arcsin(u))/d(x)=1/√(,1−u2)⋅d(u)/d(x)

3. Differentiate the inner function u=x−1 with respect to x which gives d(x−1)/d(x)=1

4. Substitute u and d(u)/d(x) back into the chain rule formula.

d(arcsin(x−1))/d(x)=1/√(,1−(x−1)2)⋅1

5. Expand the expression inside the square root.

(x−1)2=x2−2*x+1

6. Simplify the radicand by subtracting the expanded expression from 1.

1−(x2−2*x+1)=1−x2+2*x−1

1−x2+2*x−1=2*x−x2

Final Answer

d(arcsin(x−1))/d(x)=1/√(,2*x−x2)

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