Derivatives Of inverse functions; Derivatives and integrals involving exponential fun

**homerb** · February 10th 2009, 09:38 PM

Show that
a) Y= x exp-x statisfies the equation xy'=(1-x)y

**o_O** · February 10th 2009, 09:42 PM

Use product rule to get: $y = xe^{-x} \ \ \Rightarrow \ \ y' = e^{-x} - xe^{-x}$

So: $\text{LHS} = xy' = x \left(e^{-x} - xe^{-x}\right) = \cdots$

And: $\text{RHS} = (1-x)y = (1-x)(xe^{-x}) = \cdots$

Are they equal? If so, you're done.

**homerb** · February 10th 2009, 09:50 PM

do I have to use ln?

or what is the next step

**o_O** · February 10th 2009, 09:53 PM

What do you mean? We just want to show $xy' = (1-x)y$ .

We've got an expression for $xy'$ and one for $(1-x)y$ . Show that they're equal by simplifying and you're done.

**homerb** · February 10th 2009, 09:54 PM

oh I see thanks, I thinking something else?

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