Find the Derivative - d/dx y=x^(sin(x))

Problem

d()/d(x)*xsin(x)

Solution

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

ln(y)=ln(xsin(x))

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

ln(y)=sin(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(sin(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.

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

5. Evaluate the derivatives of the basic functions.

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

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

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

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

d(y)/d(x)=xsin(x)*(sin(x)/x+cos(x)*ln(x))

Final Answer

d(xsin(x))/d(x)=xsin(x)*(sin(x)/x+cos(x)*ln(x))

Want more problems? Check here!