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

Problem

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

Solution

1. Identify the rule needed for the expression, which is a product of two functions: u=x and v=sin(1/x)

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

3. Differentiate the first part u=x which gives d(x)/d(x)=1

4. Differentiate the second part v=sin(1/x) using the chain rule.

5. Apply the chain rule to v by taking the derivative of the outer function sin(u) and multiplying by the derivative of the inner function 1/x

6. Calculate the derivative of the inner function d()/d(x)1/x=−x(−2)=−1/(x2)

7. Combine the results of the chain rule to find d(sin(1/x))/d(x)=cos(1/x)⋅(−1/(x2))

8. Substitute these components back into the product rule formula.

x⋅(−1/(x2)*cos(1/x))+sin(1/x)⋅1

9. Simplify the expression by canceling x in the first term.

Final Answer

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

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