Find the Derivative - d/dx arcsin(x)

Problem

d(arcsin(x))/d(x)

Solution

1. Set up the equation by letting y=arcsin(x) which implies x=sin(y) for y in the interval [−π/2,π/2]

2. Differentiate implicitly both sides of x=sin(y) with respect to x

d(x)/d(x)=d(sin(y))/d(x)

3. Apply the chain rule to the right side of the equation.

1=cos(y)d(y)/d(x)

4. Solve for the derivative d(y)/d(x) by dividing both sides by cos(y)

d(y)/d(x)=1/cos(y)

5. Use the Pythagorean identity sin2(y)+cos2(y)=1 to express cos(y) in terms of sin(y)

cos(y)=√(,1−sin2(y))

6. Substitute x back into the expression, noting that sin(y)=x and cos(y) is non-negative on the interval [−π/2,π/2]

d(y)/d(x)=1/√(,1−x2)

Final Answer

d(arcsin(x))/d(x)=1/√(,1−x2)

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