Find the Derivative - d/dx y=arcsin(x/2)

Problem

d()/d(x)*arcsin(x/2)

Solution

1. Identify the outer function as arcsin(u) and the inner function as u=x/2

2. Apply the chain rule which states that d(y)/d(x)=d(y)/d(u)⋅d(u)/d(x)

3. Differentiate the outer function using the rule d(arcsin(u))/d(u)=1/√(,1−u2)

4. Differentiate the inner function d()/d(x)x/2=1/2

5. Combine the derivatives using the chain rule.

d(y)/d(x)=1/√(,1−(x/2)2)⋅1/2

6. Simplify the expression inside the square root.

d(y)/d(x)=1/√(,1−(x2)/4)⋅1/2

7. Find a common denominator inside the square root.

d(y)/d(x)=1/√(,(4−x2)/4)⋅1/2

8. Simplify the radical by taking the square root of the denominator.

d(y)/d(x)=1/√(,4−x2)/2⋅1/2

9. Multiply the fractions to reach the final simplified form.

d(y)/d(x)=2/√(,4−x2)⋅1/2

Final Answer

d(arcsin(x/2))/d(x)=1/√(,4−x2)

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