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

Problem

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

Solution

1. Identify the rule needed for the expression, which is the product rule for u⋅v where u=x and v=sin(2/x)

2. Apply the product rule formula d(u*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

3. Differentiate the first part u=x to get d(x)/d(x)=1

4. Differentiate the second part v=sin(2/x) using the chain rule, where the outer function is sin(u) and the inner function is 2/x

5. Calculate the derivative of the inner function d()/d(x)2/x=−2*x(−2)=−2/(x2)

6. Combine the chain rule results to find d(v)/d(x)=cos(2/x)⋅(−2/(x2))

7. Substitute all components back into the product rule expression.

x*(cos(2/x)⋅(−2/(x2)))+sin(2/x)⋅1

8. Simplify the resulting expression by canceling the x terms.

−2/x*cos(2/x)+sin(2/x)

Final Answer

d()/d(x)*x*sin(2/x)=sin(2/x)−2/x*cos(2/x)

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