Find the derivative of x^(2)e^(1/x)

How would you find the derivative of x^(2)*e^(1/x)?
Specifically, I get stuck when it comes to the derivative of e^(1/x)...
Apparently the answer is (2x-1)*e^(1/x)...
Please show working/explain?

f(x) = x^2*e((1/x)

We employ the product rule of differentiation.

df(x)/dx = (2x)*(e^(1/x)) + x^2*(-1)*(1/x^2)*e^(1/x)

The second term is obtained by employing the chain rule on e^(1/x).

Simplifying,

f '(x) = (2x)*(e^(1/x)) - e^(1/x)
f '(x) = (2x-1)*e^(1/x)

As stated.

First, use the product rule.

d/dx(x^[2])*e^(1/x)+x^(2)*d/dx(e^[1/x]…

Now take the derivative. Remember to use the chain rule with e^(1/x). Another way to think of it is e^(x^-1), so the derivative of the "inner" term is -1x^(-2).

2x*e^(1/x)+x^(2)*e^(1/x)*(-1x^-(2))

Rearranging so it's a bit clearer.

2x*e^(1/x)+x^(2)*(-1/x^(2))*e^(1/x)
2x*e^(1/x)-1*e^(1/x)
(2x-1)(e^(1/x))

The apparent answer is correct. Procedure is given in the image below, with prettyprint.

http://img23.imageshack.us/img23/1335/et…

the final derivative simplifies to the answer you supply, when factored.

[x^(2)*e^(1/x)]' = e^(1/x)[(x^2)' + x^2 (1/x)'] = (2x-1)*e^(1/x)