write the composite function in the form f(g(x)). y = e^7√x u = g(x) = 7√x y = f(u) = then find the…

Explanation:

Step1: Identify the outer - function

Given $y = e^{7\sqrt{x}}$ and $u = g(x)=7\sqrt{x}$, the outer - function $y = f(u)$ is $y = e^{u}$.

Step2: Apply the chain - rule

The chain - rule states that $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. First, find $\frac{dy}{du}$: If $y = e^{u}$, then $\frac{dy}{du}=e^{u}$. Second, find $\frac{du}{dx}$: If $u = 7\sqrt{x}=7x^{\frac{1}{2}}$, then $\frac{du}{dx}=7\times\frac{1}{2}x^{-\frac{1}{2}}=\frac{7}{2\sqrt{x}}$.

Step3: Calculate $\frac{dy}{dx}$

Substitute $u = 7\sqrt{x}$ back into $\frac{dy}{du}$ and multiply by $\frac{du}{dx}$: $\frac{dy}{dx}=e^{u}\cdot\frac{7}{2\sqrt{x}}$. Since $u = 7\sqrt{x}$, we have $\frac{dy}{dx}=e^{7\sqrt{x}}\cdot\frac{7}{2\sqrt{x}}=\frac{7e^{7\sqrt{x}}}{2\sqrt{x}}$.

Answer:

$y = f(u)=e^{u}$, $\frac{dy}{dx}=\frac{7e^{7\sqrt{x}}}{2\sqrt{x}}$