Find the Derivative - d/dx x^2sin(1/x)

Problem

d()/d(x)*x2*sin(1/x)

Solution

1. Identify the product rule, which states that d()/d(x)*ƒ(x)*g(x)=ƒ(x)′*g(x)+ƒ(x)*g(x)′ where ƒ(x)=x2 and g(x)=sin(1/x)

2. Differentiate the first part of the product, ƒ(x)=x2 using the power rule.

d(x2)/d(x)=2*x

3. Differentiate the second part of the product, g(x)=sin(1/x) using the chain rule.

d(sin(1/x))/d(x)=cos(1/x)⋅d()/d(x)1/x

4. Apply the power rule to find the derivative of the inner function 1/x=x(−1)

d()/d(x)1/x=−x(−2)=−1/(x2)

5. Combine the results of the chain rule for the second part.

d(sin(1/x))/d(x)=−1/(x2)*cos(1/x)

6. Substitute all components back into the product rule formula.

d()/d(x)*x2*sin(1/x)=2*x*sin(1/x)+x2*(−1/(x2)*cos(1/x))

7. Simplify the expression by canceling the x2 terms in the second part.

d()/d(x)*x2*sin(1/x)=2*x*sin(1/x)−cos(1/x)

Final Answer

d()/d(x)*x2*sin(1/x)=2*x*sin(1/x)−cos(1/x)

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