Question

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=(6x + 5)^8
write y=(6x + 5)^8 in the form y = f(u) and u = g(x). choose the correct functions f(u) and g(x) below.
a. f(u)=(6u + 5)^8
g(x)=6x
b. f(u)=6u + 5
g(x)=x^8
c. f(u)=u^8
g(x)=6x + 5
d. f(u)=6u^8
g(x)=x + 5

Explanation:

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

We want to express $y=(6x + 5)^8$ as a composition of functions. Let $u = 6x+5$, so $g(x)=6x + 5$, and $y = u^8$, so $f(u)=u^8$. So the correct choice is C.

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

Differentiate $u = g(x)=6x + 5$ with respect to $x$. Using the power - rule, $\frac{du}{dx}=\frac{d}{dx}(6x + 5)=6$.

Step3: Find $\frac{dy}{du}$

Differentiate $y = f(u)=u^8$ with respect to $u$. Using the power - rule, $\frac{dy}{du}=8u^7$.

Step4: Use the chain - rule

The chain - rule states that $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substitute $\frac{dy}{du}=8u^7$ and $\frac{du}{dx}=6$ into the chain - rule formula. Then substitute $u = 6x+5$ back in. We get $\frac{dy}{dx}=8u^7\cdot6=48(6x + 5)^7$.

Answer:

C. $f(u)=u^8$, $g(x)=6x + 5$; $\frac{dy}{dx}=48(6x + 5)^7$