# Integrate AFTER differentiate exponent to power exponent

#### sjaffry

Hi all,

I need to differentiate the following

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

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

#### hollywood

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