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

Problem

d()/d(x)*x^2*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)=x^2 and g(x)=sin(1/x)

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

d(x^2)/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/(x^2)

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

d(sin(1/x))/d(x)=−1/(x^2)*cos(1/x)

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

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

7. Simplify the expression by canceling the x^2 terms in the second part.

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

Final Answer

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

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