Question

question part 1 of 2 completed: 4 of 20 | my score: 4/20 pts (20%) write the function below in the form y = f(u) and u = g(x), then find \\(\frac{dy}{dx}\\) as a function of x. y=(3x + 4)^7 write y=(3x + 4)^7 in the form y = f(u) and u = g(x). choose the correct functions f(u) and g(x) below. a. f(u)=u^7 g(x)=3x + 4 b. f(u)=3u + 4 g(x)=x^7 c. f(u)=(3u + 4)^7 g(x)=3x d. f(u)=3u^7 g(x)=x + 4

Explanation:

Step1: Identify f(u) and g(x)

For the composite - function $y=(3x + 4)^7$, if we let $u = 3x+4$ and $y = u^7$. Then $f(u)=u^7$ and $g(x)=3x + 4$.

Step2: Find $\frac{dy}{du}$ and $\frac{du}{dx}$

Differentiate $y = f(u)=u^7$ with respect to $u$. Using the power - rule $\frac{d}{du}(u^n)=nu^{n - 1}$, we get $\frac{dy}{du}=7u^6$.
Differentiate $u = g(x)=3x + 4$ with respect to $x$. Using the sum - rule and the power - rule, $\frac{du}{dx}=3$.

Step3: Apply the chain - rule

The chain - rule states that $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$.
Substitute $\frac{dy}{du}=7u^6$ and $\frac{du}{dx}=3$ into the chain - rule formula. Since $u = 3x+4$, we have $\frac{dy}{dx}=7(3x + 4)^6\cdot3=21(3x + 4)^6$.

For the multiple - choice part:

Answer:

A. $f(u)=u^7$, $g(x)=3x + 4$
$\frac{dy}{dx}=21(3x + 4)^6$