Single Variable Calculus, Chapter 3, 3.5, Section 3.5, Problem 2

Express the composite function $y = \sqrt{4+3x}$ in the form $f(g(x))$. [Identify the inner function $u=g(x)$ and the outer function $y = f(u)$.] Then find the derivative $\displaystyle \frac{dy}{dx}$

Let $y = f(g(x))$ where $u = g(x) = 4 + 3x$ and $y = f(u) = \sqrt{u}$

Then


$
\begin{equation}
\begin{aligned}
y' &= \frac{dy}{dx} = \frac{dy}{du} \frac{du}{dx}\\
\\
y' &= \frac{d}{du} (\sqrt{u}) \cdot \frac{d}{dx} (4 + 3x)\\
\\
y' &= \left( \frac{1}{2 \sqrt{u}} \right) (3) && \text{Simplify the equation}\\
\\
y' &= \frac{3}{2\sqrt{u}} && \text{Substitute the value of } u \\
\\
y' &= \frac{3}{2 \sqrt{4+3x}}
\end{aligned}
\end{equation}
$