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 = (2x + 9)^8

Explanation:

Step1: Identify inner and outer functions

Let $u = g(x)=2x + 9$ and $y = f(u)=u^{8}$.

Step2: Find the derivative of the inner function

The derivative of $u = g(x)=2x + 9$ with respect to $x$ is $g^\prime(x)=\frac{du}{dx}=2$.

Step3: Find the derivative of the outer function

The derivative of $y = f(u)=u^{8}$ with respect to $u$ is $f^\prime(u)=\frac{dy}{du}=8u^{7}$.

Step4: Apply the chain - rule

By the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substitute $u = 2x + 9$, $\frac{dy}{du}=8u^{7}$ and $\frac{du}{dx}=2$ into the chain - rule formula. We get $\frac{dy}{dx}=8(2x + 9)^{7}\cdot2$.

Step5: Simplify the result

$\frac{dy}{dx}=16(2x + 9)^{7}$.

Answer:

$\frac{dy}{dx}=16(2x + 9)^{7}$