Question

Write the composite function in the form f(g(x)). [Identify the inner function u = g(x) and the outer function y = f(u).] $ y = e^{{\color{red}5}\sqrt{x}} $

Answer 1

Answer:

The answer is "$\frac{5 e^{5\sqrt{x} }}{2\sqrt{x}}$".

Step-by-step explanation:

Given:

$y = e^{{\color{\red}5}\sqrt{x}}$

let

$\to t= 5\sqrt{x}\\\\\frac{dt}{dx}= 5 \frac{1}{2\sqrt{x}}\\\\\frac{dt}{dx}= \frac{5}{2\sqrt{x}}$

and

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

$\to \frac{dy}{dt}=e^{5\sqrt{x} }$

So,

$\to \frac{dy}{dx}= \frac{dy}{dt} \times \frac{dt}{dx}$

         $=e^{5\sqrt{x} }\times \frac{5}{2\sqrt{x}}\\\\= \frac{5 e^{5\sqrt{x} }}{2\sqrt{x}}$

OR

$\to g(x) = 5\sqrt{x} \\\\\to f(x) = e^{(x)}$

Derivate:  

$\to f''g' = \frac{e^{(5\sqrt{x})}5}{(2\sqrt{x})}$