# Derivative problem

## Homework Statement

I actually know the solution, but cannot reach it on paper:

Find where the slope of the tangent to the curve e^(-x^2) is equal to 2/e

## The Attempt at a Solution

d/dx e^(-x^2) = e^(-x^2) * d/dx -x^2 = -2xe^(-x^2)

Set: -2xe^(-x^2) = 2/e

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

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

e^(x^2-1) = -x

At this point I can see that the only possible solution is x = -1, yet cannot actually reach that conclusion on paper. It seems like I'm missing something ridiculously simple.

Any help greatly appreciated!

AKG
Homework Helper
What do you mean you can't reach that conclusion on paper? Plug in x = -1, show that left sides = right side, then all that remains to show is that there are no more solutions. The function mapping x to ex²-1+x is clearly increasing, so it has at most one root, and you've found it at -1, so you're done.

I mean that I know that the solution is x = -1 because I just happened to notice that it was the solution, not because I solved for x = -1.

Say I'm left with:
e^(x^2-1) = -x

I could easily have no idea what the solution is.
How do I work the equation to show x = -1?

rock.freak667
Homework Helper
sub x=1 into the equation and show the left side is equal to the right side

This is a problem that, as far as I know, can't be solved algebraically. You'd have to use analysis as explained above.

HallsofIvy
"Happening to notice" that a specific value satisfies an equation is a perfectly good method! (Provided that you check that it does work.) And a perfectly good method of seeing that x= -1 is the ONLY solution is to graph the functions $$y= e^{x^2-1}$$ and $$y= -x$$.