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

Problem

d()/d(x)*x^2*e^(−1/x)

Solution

1. Identify the rule needed for the derivative of a product of two functions, u=x^2 and v=e^(−1/x)

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

3. Differentiate the first part, u=x^2 using the power rule to get d(x^2)/d(x)=2*x

4. Differentiate the second part, v=e^(−1/x) using the chain rule.

5. Calculate the derivative of the exponent, d()/d(x)−x^(−1)=x^(−2)=1/(x^2)

6. Combine the chain rule results to find d(e^(−1/x))/d(x)=e^(−1/x)⋅1/(x^2)

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

d()/d(x)*x^2*e^(−1/x)=x^2⋅(e^(−1/x)⋅1/(x^2))+e^(−1/x)⋅(2*x)

8. Simplify the expression by canceling x^2 in the first term and factoring out common terms.

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

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

Final Answer

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

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