Find the Derivative - d/dx arccos(x)

Problem

d(arccos(x))/d(x)

Solution

1. Set up the equation by letting y=arccos(x) which implies that cos(y)=x for y in the interval [0,π]

2. Differentiate implicitly both sides of the equation cos(y)=x with respect to x

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

−sin(y)d(y)/d(x)=1

3. Solve for the derivative d(y)/d(x) by dividing both sides by −sin(y)

d(y)/d(x)=−1/sin(y)

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

sin(y)=√(,1−cos2(y))

5. Substitute cos(y)=x back into the expression for the derivative.

d(y)/d(x)=−1/√(,1−x2)

Final Answer

d(arccos(x))/d(x)=−1/√(,1−x2)

Want more problems? Check here!