Differentiate with e

Question

babynini · Answer

$$y=\frac{ ae^x+b }{ ce^x+d }$$y'=?

tkhunny · Answer

Have we attempted the Quotient Rule?  What has "e" to do with it?

babynini · Answer

$$\frac{ (ce^x+d)(ae^x+b)'-(ae^x+b)(ce^x+d)' }{ (ce^x+d)^2 }$$erm yeah, I did this. But i'm not sure how to differentiate with those a's and b's and e's.
E becaue of the e^x's in there.

babynini · Answer

I mean, are c,d,b, a all constants? do they become 1?

tkhunny · Answer

That';s good.  $(ae^{x}+b)' = ae^{x}$  a, b, c, d, and e are just constants.  e is a specific constant.  The others are arbitrary.

tkhunny · Answer

$\dfrac{d}{dx}a = 0$

$\dfrac{d}{dx}e = 0$

$\dfrac{d}{dx}a^{x} = a^{x}\ln(a)$

$\dfrac{d}{dx}e^{x} = e^{x}\ln(e) = e^{x}$

babynini · Answer

hmm ok.
so in ae^x the a stays there because it is.."attached" to e?

tkhunny · Answer

It's just a constant and comes along for the ride.

tkhunny · Answer

$\dfrac{d}{dx}ae^{x} = a\dfrac{d}{dx}e^{x} = ae^{x}$

tkhunny · Answer

$\dfrac{d}{dx}(ae)^{x} = (ae)^{x}\ln(ae) = (ae)^{x}(\ln(a)+1)$

tkhunny · Answer

Try a few examples and get used to what is a constant and what isn't.

babynini · Answer

Ohh I see! That makes more sense written that way with the a before the d/dx

babynini · Answer

$$\frac{ e^x(ad-bc) }{ (c^ex+d)^2}$$=y'

babynini · Answer

oops that botton was meant to be
ce^x

tkhunny · Answer

I think you have it!  Good work.

Very often, there will be some odd simplification with the denominator.  I don't see any, this time.  Keep your eyes out for it.

babynini · Answer

Thanks so much! :)

babynini · Answer

ah ok. In this one there was just a lot of multiplying in and then taking back out again xD