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
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 $\displaystyle x = 0$.
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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.