Question

for the following composite function, find an inner function $u = g(x)$ and an outer function $y = f(u)$ such that $y = f(g(x))$. then calculate $\frac{dy}{dx}$ $y=(3x + 7)^{11}$ identify the inner and outer functions. choose the correct answer. a. $u = g(x)=3x + 7$ and $y = f(u)=u^{11}$ b. $u = g(x)=3x + 7$ and $y = f(u)=u$ c. $u = g(x)=3x$ and $y = f(u)=u^{11}$ d. $u = g(x)=x^{11}$ and $y = f(u)=3u + 7$ $\frac{dy}{dx}=\square$

Explanation:

Step1: Identify inner and outer functions

For \(y=(3x + 7)^{11}\), if we let \(u = g(x)=3x+7\) (the inner - function, a linear function), then \(y = f(u)=u^{11}\) (the outer - function, a power function). So the correct option is A.

Step2: Use the chain rule

The chain rule states that \(\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}\).

• First, find \(\frac{dy}{du}\):

Since \(y = u^{11}\), by the power rule \(\frac{d}{du}(u^{n})=nu^{n - 1}\), we have \(\frac{dy}{du}=11u^{10}\).

• Then, find \(\frac{du}{dx}\):

Since \(u = 3x+7\), \(\frac{du}{dx}=3\).

Step3: Calculate \(\frac{dy}{dx}\)

Substitute \(u = 3x + 7\) into \(\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}\).
\(\frac{dy}{dx}=11u^{10}\cdot3\).
Replace \(u\) with \(3x + 7\), we get \(\frac{dy}{dx}=11(3x + 7)^{10}\cdot3\).
Simplify the expression: \(\frac{dy}{dx}=33(3x + 7)^{10}\).

Answer:

A. \(u = g(x)=3x + 7\) and \(y = f(u)=u^{11}\); \(\frac{dy}{dx}=33(3x + 7)^{10}\)