Find the Derivative - d/d@VAR f(x)=x^(cos(x))

Problem

d()/d(x)*xcos(x)

Solution

1. Identify the function as a power-exponential form y=xcos(x) which requires logarithmic differentiation.

2. Apply the natural logarithm to both sides of the equation y=xcos(x) to simplify the exponent.

ln(y)=ln(xcos(x))

3. Use the power rule for logarithms to move the exponent in front of the log.

ln(y)=cos(x)*ln(x)

4. Differentiate implicitly both sides with respect to x applying the product rule to the right side.

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

5. Apply the product rule (d(u)*v)/d(x)=u′*v+u*v′ to the right side.

1/yd(y)/d(x)=−sin(x)*ln(x)+cos(x)1/x

6. Solve for the derivative d(y)/d(x) by multiplying both sides by y

d(y)/d(x)=y*(cos(x)/x−sin(x)*ln(x))

7. Substitute the original expression for y back into the equation.

d(y)/d(x)=xcos(x)*(cos(x)/x−sin(x)*ln(x))

Final Answer

d(xcos(x))/d(x)=xcos(x)*(cos(x)/x−sin(x)*ln(x))

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