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
\\(\frac{dy}{dx}=\square\\)

Explanation:

Step1: Identify inner - outer functions

We have $y = f(u)$ and $u = g(x)$. Given $y=(6x + 5)^8$, if we let $u = 6x+5$ and $y = u^8$, then $f(u)=u^8$ and $g(x)=6x + 5$.

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

Differentiate $y = f(u)=u^8$ with respect to $u$ using the power - rule $\frac{d}{du}(u^n)=nu^{n - 1}$. So, $\frac{dy}{du}=8u^{7}$.

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

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

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. So, $\frac{dy}{dx}=8(6x + 5)^{7}\cdot6$.

Step5: Simplify the result

$\frac{dy}{dx}=48(6x + 5)^{7}$.

Answer:

$48(6x + 5)^{7}$