Integrate AFTER differentiate exponent to power exponent

Jun 2017
37
0
New Zealand
Hi all,

I need to differentiate the following

\begin{equation}
f(t) = \frac{\partial }{\partial t}e^{-ne^{-\lambda t}}
\end{equation}

$$\int_{0}^{+Inf} t dt = ?$$
 
Mar 2010
1,055
290
To take the derivative in the first equation, use the chain rule:

\(\displaystyle \frac{\partial}{\partial t}e^{-ne^{-\lambda t}}=\frac{\partial}{\partial t}e^{-n(e^{-\lambda t})}=-ne^{-ne^{-\lambda t}}\frac{\partial}{\partial t}e^{-\lambda t}=n\lambda e^{-ne^{-\lambda{t}}}e^{-\lambda t}\)

The integral diverges to infinity:

\(\displaystyle \int_0^\infty t\,dt=\lim_{s\to\infty}\int_0^s t\,dt=\lim_{s\to\infty}\frac{1}{2}s^2=\infty\)

- Hollywood