# Finding the derivative of an equation with an exponent

1. Oct 13, 2007

### fk378

1. The problem statement, all variables and given/known data
Find the local max and min
f(x)=xe^x

2. Relevant equations
first find the derivative of xe^x and then set it to 0

3. The attempt at a solution
I understand how to approach the problem but I don't know how to break apart the original equation since there is an exponent there. I know if I ln both sides I can get rid of the e and therefore bring down the x exponent, but ln(0) does not exist.

2. Oct 13, 2007

### bob1182006

to find the derivative you need to use the product rule:

(uv)'=uv'+vu'

and from there you can find the critical points.

3. Oct 13, 2007

### HallsofIvy

Staff Emeritus
(xex)'= (x)' ex+ x(ex)'

Now, are you saying that you do not know the derivative of ex?

4. Oct 13, 2007

### fk378

xe^x
f'(x)= (e^x) (x)(e^x)
to find the critical points:
(e^x) (x)(e^x)=x(e^2x) = 0

Now how do I solve for x when there is an x in the exponent?

5. Oct 13, 2007

### nrqed

You made a mistake. You must have a *sum* of two terms, not a product.
So you must solve $e^x + x e^x =0$ Factor out the exponential and find when this will be equal to zero.

Last edited: Oct 13, 2007
6. Oct 13, 2007

### fk378

So then I have e^x (1+x) = 0. How do I factor out the x when there is one in the exponent? I know I can take the natural log but can I do that for the 0 as well? but ln(0) does not exist.

Last edited: Oct 13, 2007
7. Oct 13, 2007

### rock.freak667

No need to take ln...if you know that e^x is never equal to zero for all values of x, then you can simply divide by e^x

8. Oct 13, 2007

You have a product with two factors; only one of them has to equal zero in order for the entire product to equal zero.

9. Oct 13, 2007

### atqamar

Here is another way to solve for the derivitive. It is similar to the product rule, but I like it's elegance. And it has to do with the fact that $$\frac{d}{dx}ln(x)=\frac{1}{x}$$.

$$f(x)=xe^x$$

Take natural log of both sides:
$$ln(f(x))=ln(x)+x$$

Use the chain rule to find the derivitive of the left side, and then differentiate the right:
$$\frac{f'(x)}{f(x)}=\frac{1}{x}+1$$

Since $$f(x)$$ is given above, multiply both sides by it and you end up with:
$$f'(x)=e^x+xe^x$$

Now when you set the derivitive to $$0$$, you factor out the $$e^x$$ and use the zero product rule along with some function analysis to get your solution. Remember the analysis!
$$0=e^x(x+1)$$.

10. Oct 13, 2007

### nrqed

You caNOT isolate x in that expression. So what youhave to do is to say the following: the product is zero if either of the two factors is zero. So you turn it into two separate problems. SI there any x for which $e^x=0$ and is there any x for which $(1+x) =0$ ?

11. Oct 13, 2007

### HallsofIvy

Staff Emeritus

12. Oct 13, 2007