write the composite function in the form f(g(x)). identify the inner function u = g(x) and the outer…

Explanation:

Step1: Identify inner and outer functions

For $y = \tan(x^{9})$, let $u = g(x)=x^{9}$ and $y = f(u)=\tan(u)$. So $(f(u),g(x))=(\tan(u),x^{9})$.

Step2: Apply chain - rule

The chain - rule states that $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. The derivative of $y = f(u)=\tan(u)$ with respect to $u$ is $\frac{dy}{du}=\sec^{2}(u)$. The derivative of $u = g(x)=x^{9}$ with respect to $x$ is $\frac{du}{dx}=9x^{8}$.

Step3: Substitute $u$ back and find $\frac{dy}{dx}$

Substitute $u = x^{9}$ into $\frac{dy}{du}\cdot\frac{du}{dx}$. We get $\frac{dy}{dx}=\sec^{2}(x^{9})\cdot9x^{8}=9x^{8}\sec^{2}(x^{9})$.

Answer:

$(f(u),g(x))=(\tan(u),x^{9})$ $\frac{dy}{dx}=9x^{8}\sec^{2}(x^{9})$