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

Problem

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

Solution

1. Identify the function to be differentiated, which is the inverse sine function y=arcsin(x)

2. Rewrite the equation in terms of x by taking the sine of both sides, resulting in sin(y)=x

3. Differentiate implicitly with respect to x on both sides of the equation.

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

4. Apply the chain rule to the left side and the power rule to the right side.

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

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

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

6. Substitute back in terms of x using the Pythagorean identity cos(y)=√(,1−sin2(y))

d(y)/d(x)=1/√(,1−sin2(y))

7. Simplify the expression by replacing sin(y) with x

d(y)/d(x)=1/√(,1−x2)

Final Answer

d(arcsin(x))/d(x)=1/√(,1−x2)

Want more problems? Check here!