Find dy/dx y=x^(cos(x))

Problem

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

Solution

1. Apply logarithmic differentiation by taking the natural logarithm of both sides to handle the variable in the exponent.

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

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

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

3. Differentiate both sides with respect to x using the chain rule on the left side and the product rule on the right side.

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

4. Apply the product rule (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x) to the right side of the equation.

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

5. Compute the derivatives of the individual functions ln(x) and cos(x)

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

6. Solve for dy/dx 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 to get the final derivative in terms of x

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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