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

Problem

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

Solution

1. Identify the inner and outer functions to prepare for the chain rule, where the outer function is cos(u) and the inner function is u=arcsin(x)

2. Apply the chain rule by multiplying the derivative of the outer function with respect to the inner function by the derivative of the inner function.

d(cos(arcsin(x)))/d(x)=−sin(arcsin(x))⋅d(arcsin(x))/d(x)

3. Simplify the trigonometric composition sin(arcsin(x)) which equals x for the domain of the inverse sine function.

−sin(arcsin(x))=−x

4. Differentiate the inverse sine function using the standard formula d(arcsin(x))/d(x)=1/√(,1−x2)

d(cos(arcsin(x)))/d(x)=−x⋅1/√(,1−x2)

5. Combine the terms into a single fraction to reach the final derivative.

d(cos(arcsin(x)))/d(x)=−x/√(,1−x2)

Final Answer

d(cos(arcsin(x)))/d(x)=−x/√(,1−x2)

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