# Thread: Integrate AFTER differentiate exponent to power exponent

1. ## Integrate AFTER differentiate exponent to power exponent

Hi all,

I need to differentiate the following

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

$$\int_{0}^{+Inf} t dt = ?$$

2. ## Re: Integrate AFTER differentiate exponent to power exponent

To take the derivative in the first equation, use the chain rule:

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

The integral diverges to infinity:

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

- Hollywood