find the derivative of (10,000)/(9+4e^(-.2t))

Question

solomonzelman · Answer

with respect to what ?

solomonzelman · Answer

if it is not with respect to 'e' then you are differentiating a constant.

anonymous · Answer

it's actually e^-.2t sorry

solomonzelman · Answer

$\Large\color{black}{ \displaystyle  y=e^{-.2t}}$

solomonzelman · Answer

well, you can just do the derivative of e^t is e^t, but then use the chain rule for the exponent.

anonymous · Answer

$$\frac{ 10,000 }{ 9+4e ^{-.2t} }$$

solomonzelman · Answer

$\Large\color{black}{ \displaystyle  y=10,000\left(9+4e^{-.2t}\right)^{-1}}$ this is how I would write it before differentiating

solomonzelman · Answer

$\large\color{black}{ \displaystyle  y'=10,000(-1)\left(9+4e^{-.2t}\right)^{-1-1}~\times~\frac{d}{dx}\left[9+4e^{-2.t}\right]}$

solomonzelman · Answer

or, if you want you can do the quotient rule, if you feel that is easier

solomonzelman · Answer

maybe lets differentiate first, the
$\large\color{black}{ \displaystyle  \left(9+4e^{-.2t}\right)^{-1}}$

anonymous · Answer

so i did it the way you said to by differentiating and it was way easier than the quotient rule

solomonzelman · Answer

what did you get?

anonymous · Answer

i figured out the answer. i think the quotient rule was just confusing me.

anonymous · Answer

in the end i got $$\frac{ 80,000e ^{-.2t} }{ 9+4e ^{-.2t} }$$

solomonzelman · Answer

I think the bottom should be squared, because you are applying the power rule to
$\large\color{black}{ \displaystyle  \left(9+4e^{-.2t}\right)^{-1}}$

and the 80000 should be 8000, as you are multiplying the 10000 coefficient times 0.2

and you are right that it is positive, because negative that is from the exponent's chain rule,
and the negative that is from the power rule\
CANCEL each other out.

solomonzelman · Answer

$\large\color{black}{ \displaystyle  y'=10,000(-1)\left(9+4e^{-.2t}\right)^{-1-1}~\times~\frac{d}{dx}\left[9+4e^{-2.t}\right]}$

$\large\color{black}{ \displaystyle  y'=10,000(-1)\left(9+4e^{-.2t}\right)^{-2}~\times~\left[4e^{-2.t}\right] \times -.2t}$

and on...

anonymous · Answer

cool thank you :)

solomonzelman · Answer

yes, .2 times 4 = 0.8
amd 10000 times 0.8 = 8000

solomonzelman · Answer

that is why the coefficient in the answer is 8000

solomonzelman · Answer

bye