1. derivatives of e functions

a little confused on this problem, any step by step help is appreciated.

find the derivative and f(0)

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

2. Originally Posted by JGaraffa
f(x) = e^(x^2 + 2) / e^(x^2) + 1
Is it $\displaystyle f(x) = \dfrac{e^{x^2 + 2}}{ e^{x^2+ 1}}$, or $\displaystyle f(x) = \dfrac{e^{x^2 + 2}}{ e^{x^2}+ 1}$, or $\displaystyle f(x) = \dfrac{e^{x^2 + 2}}{ e^{x^2}}+1$? When posting a question, your priority must be on clarity.

3. 2nd one. sorry i don't really know how to use the symbols.

4. Originally Posted by JGaraffa
a little confused on this problem, any step by step help is appreciated.

find the derivative and f(0)

f(x) = e^(x^2 + 2) / [e^(x^2) + 1]
quotient rule ...

$\displaystyle \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \cdot u' - u \cdot v'}{v^2}$

$\displaystyle f'(x) = \frac{(e^{x^2}+1)e^{x^2+2} \cdot 2x - e^{x^2+2} \cdot e^{x^2} \cdot 2x}{(e^{x^2}+1)^2}$

evaluate $\displaystyle f'(0)$

5. Originally Posted by JGaraffa
2nd one. sorry i don't really know how to use the symbols.
If $\displaystyle f(x) = \dfrac{e^{x^2 + 2}}{ e^{x^2}+ 1}$, then, using the quotient rule, $\displaystyle f'(x) = \dfrac{\left(e^{x^2}+ 1\right)\dfrac{d}{dx}\left(e^{x^2 + 2}\right)-\left(e^{x^2 + 2}\right)\dfrac{d}{dx}\left(e^{x^2}+ 1\right)}{\left(e^{x^2}+1\right)^2}$. By the Chain Rule, $\displaystyle \dfrac{d}{dx}\left(e^{x^2 + 2}\right) = 2xe^{x^2+2}$ and $\displaystyle \dfrac{d}{dx}\left(e^{x^2}+ 1\right) = 2e^{x^2}$. So $\displaystyle f'(x) = \dfrac{\left(e^{x^2}+ 1\right)\left(2xe^{x^2+2}\right)-\left(e^{x^2 + 1}\right)\left(2e^{x^2}\right)}{\left(e^{x^2}+1\ri ght)^2}$. Simplifying this gives $\displaystyle f'(x)= \boxed{\dfrac{2e^{x^2+2}x}{\left(e^{x^2}+1\right)}- \dfrac{2e^{2 x^2+2}x}{(e^{x^2}+1)^2}}$. To find $\displaystyle f'(0)$, let $\displaystyle x = 0$.

You can learn how to use the Latex from here.

6. thanks for your help guys i really appreciate it. hopefully you're still around to check this and i won't have to post another thread for a simple question...

when solving for f`(0) what is the correct order of operations for functions of e such as the one's above. for example, for $\displaystyle e^(x^2)$ would i do $\displaystyle 0^2 * e^x$ ? cause when i do that the answer comes out to zero, and online function calculators tell me the answer should come out to 2.

same for $\displaystyle e^(x^2 + 2)$

ugh sorry i'm still really bad at this latex stuff.