Find the Derivative - d/dx y=x^(2sin(x))

Problem

d()/d(x)*x(2*sin(x))

Solution

1. Apply logarithmic differentiation by setting y=x(2*sin(x)) and taking the natural logarithm of both sides to handle the variable exponent.

ln(y)=ln(x(2*sin(x)))

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

ln(y)=2*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.

1/yd(y)/d(x)=d()/d(x)*(2*sin(x)*ln(x))

4. Apply the product rule d()/d(x)*[ƒ(x)*g(x)]=ƒ(x)′*g(x)+ƒ(x)*g(x)′ to the right side.

1/yd(y)/d(x)=2*cos(x)*ln(x)+2*sin(x)1/x

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

d(y)/d(x)=y*(2*cos(x)*ln(x)+(2*sin(x))/x)

6. Substitute the original expression for y back into the equation to get the final result in terms of x

d(y)/d(x)=x(2*sin(x))*(2*cos(x)*ln(x)+(2*sin(x))/x)

Final Answer

d(x(2*sin(x)))/d(x)=x(2*sin(x))*(2*cos(x)*ln(x)+(2*sin(x))/x)

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