# Math Help - trouble taking a derivative

1. ## trouble taking a derivative

I need help taking the following derivatives:

d/dt of (e^-2)*(e^(1+e^x))*(e^t)

and

d/dt of (e^-2)*(e^(1+e^t))*(e^x)

thanks

Fred1956

2. Originally Posted by Fred1956
I need help taking the following derivatives:

d/dt of (e^-2)*(e^(1+e^x))*(e^t)
If $x \neq f(t)$ let $u = e^{-2}(e^{1+e^x})$ which is a constant which gives:

$\frac{d}{dt}(ue^t)$ where u is a constant, should be easy enough.

d/dt of (e^-2)*(e^(1+e^t))*(e^x)
$\frac{d}{dt}(e^{-2}e^x)(e^{1+e^t})$

Let $y = (e^{-2}e^x)(e^{1+e^t})$

take logs:

Spoiler:
$ln(y) = ln[(e^{-2}e^x)(e^{1+e^t})]$

(Because $ln(abc) = ln(a)+ln(b)+ln(c)$)
$= ln(e^{-2}) + ln(e^x) + ln(e^{1+e^t})$

(Because $ln(a^k) = k\,ln(a)$ and $ln(e^k) = k$)
$= -2 + x + 1+e^t = e^t+x-1$

$ln(y) = e^t+x-1$

Differentiate implicitly:

$\frac{1}{y} \frac{dy}{dt} = e^t$

$
\frac{dy}{dt} = ye^t = e^t(e^{-2}e^x)(e^{1+e^t})$