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}7}\sqrt{x}} $
(g(x), f(u)) = ??
and find the derivative

For what values of x does the graph of f have a horizontal tangent? (Use n as your integer variable. Enter your answers as a comma-separated list.)
f(x) = x ? 2 sin x
x=??

a) (g(x), f(u)) = ( 7*√x , e^u )

b)   y ' = 3.5 * e^(7*√x) / √x

Step-by-step explanation:

Given:

- The given function:

                                       y = e^(7*√x)

Find:

- Express the given function as a composite of f(g(x)). Where, u = g(x) and y = f(u).

- Express the derivative of y, y'?

Solution:

- We will assume the exponent of  the natural log to be the u. So u is:

                                     u = g(x) = 7*√x

- Then y is a function of u as follows:

                                     y = f(u) = e^u

- The composite function is as follows:

                                    (g(x), f(u)) = ( 7*√x , e^u )

- The derivative of y is such that:

                                    y = f(g(x))

                                    y' = f' (g(x) ) * g'(x)

                                    y' = f'(u) * g'(x)

                                    y' = e^u* 3.5 / √x

- Hence,

                                   y ' = 3.5 * e^(7*√x) / √x

Write a proportion using the ratios. Find the cross products. If the cross products are equal, then the ratios form a proportion.