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

Problem

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

Solution

1. Set the expression equal to a variable y to use logarithmic differentiation.

y=xcos(x)

2. Apply the natural logarithm to both sides of the equation to move the exponent.

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

3. Use the power rule for logarithms to simplify the right side.

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

4. Differentiate implicitly with respect to x on both sides.

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

5. Apply the chain rule to the left side and the product rule to the right side.

1/yd(y)/d(x)=cos(x)d(ln(x))/d(x)+ln(x)d(cos(x))/d(x)

6. Evaluate the derivatives of the basic functions.

1/yd(y)/d(x)=cos(x)1/x+ln(x)*(−sin(x))

7. Simplify the expression on the right side.

1/yd(y)/d(x)=cos(x)/x−sin(x)*ln(x)

8. Solve for the derivative by multiplying both sides by y

d(y)/d(x)=y*(cos(x)/x−sin(x)*ln(x))

9. 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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