for the following composite function, find an inner function $u = g(x)$ and an outer function $y = f(u)$…

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 $y = f(u)=3u + 7$ $\\frac{dy}{dx}=\\square$

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 $y = f(u)=3u + 7$ $\\frac{dy}{dx}=\\square$

Answer

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})