Find the Derivative - d/dx y=xsin(1/x)

Problem

d()/d(x)*x*sin(1/x)

Solution

1. Identify the rule needed for the expression x*sin(1/x) which is the product rule: (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

2. Assign the variables for the product rule where u=x and v=sin(1/x)

3. Differentiate u with respect to x to get d(u)/d(x)=1

4. Differentiate v using the chain rule, where the outer function is sin(u) and the inner function is 1/x=x(−1)

5. Apply the chain rule to find d(v)/d(x)=cos(1/x)⋅(−x(−2))=−1/(x2)*cos(1/x)

6. Combine the parts using the product rule formula: x*(−1/(x2)*cos(1/x))+sin(1/x)*(1)

7. Simplify the resulting expression by multiplying x and −1/(x2)

Final Answer

(d(x)*sin(1/x))/d(x)=sin(1/x)−1/x*cos(1/x)

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