derivatives of e functions

**JGaraffa** · May 2nd 2010, 04:38 PM

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

**TheCoffeeMachine** · May 2nd 2010, 04:46 PM

Originally Posted by JGaraffa

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

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

**JGaraffa** · May 2nd 2010, 05:32 PM

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

**skeeter** · May 2nd 2010, 05:53 PM

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 ...

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

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

evaluate $f'(0)$

**TheCoffeeMachine** · May 2nd 2010, 06:13 PM

Originally Posted by JGaraffa

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

If $f(x) = \dfrac{e^{x^2 + 2}}{ e^{x^2}+ 1}$ , then, using the quotient rule, $f'(x) = \dfrac{\left(e^{x^2}+ 1\right)\dfrac{d}{dx}\left(e^{x^2 + 2}\right)-\left(e^{x^2 + 2}\right)\dfrac{d}{dx}\left(e^{x^2}+ 1\right)}{\left(e^{x^2}+1\right)^2}$ . By the Chain Rule, $\dfrac{d}{dx}\left(e^{x^2 + 2}\right) = 2xe^{x^2+2}$ and $\dfrac{d}{dx}\left(e^{x^2}+ 1\right) = 2e^{x^2}$ . So $f'(x) = \dfrac{\left(e^{x^2}+ 1\right)\left(2xe^{x^2+2}\right)-\left(e^{x^2 + 1}\right)\left(2e^{x^2}\right)}{\left(e^{x^2}+1\ri ght)^2}$ . Simplifying this gives $f'(x)= \boxed{\dfrac{2e^{x^2+2}x}{\left(e^{x^2}+1\right)}- \dfrac{2e^{2 x^2+2}x}{(e^{x^2}+1)^2}}$ . To find $f'(0)$ , let $x = 0$ .

You can learn how to use the Latex from here.

**JGaraffa** · May 4th 2010, 09:06 AM

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 $e^(x^2)$ would i do $0^2 * e^x$ ? 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 $e^(x^2 + 2)$

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

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