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

Problem

d(cos(x*sin(x)))/d(x)

Solution

1. Identify the outer function and the inner function to apply the Chain Rule. The outer function is cos(u) and the inner function is u=x*sin(x)

2. Apply the Chain Rule by taking the derivative of the outer function with respect to the inner function and multiplying it by the derivative of the inner function.

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

3. Apply the Product Rule to find the derivative of the inner function x*sin(x) The Product Rule states (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

(d(x)*sin(x))/d(x)=x*cos(x)+sin(x)⋅1

4. Substitute the result of the Product Rule back into the Chain Rule expression.

d(cos(x*sin(x)))/d(x)=−sin(x*sin(x))*(x*cos(x)+sin(x))

5. Distribute the negative sign to simplify the final expression.

d(cos(x*sin(x)))/d(x)=−(x*cos(x)+sin(x))*sin(x*sin(x))

Final Answer

d(cos(x*sin(x)))/d(x)=−(x*cos(x)+sin(x))*sin(x*sin(x))

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