if f(x)=e^√(2x + 7), find f(x).

Explanation:

Step1: Identify the outer - inner functions

Let $u = \sqrt{2x + 7}=(2x + 7)^{\frac{1}{2}}$, then $y = e^{u}$.

Step2: Differentiate the outer function

The derivative of $y = e^{u}$ with respect to $u$ is $\frac{dy}{du}=e^{u}$.

Step3: Differentiate the inner function

Using the power - rule, if $u=(2x + 7)^{\frac{1}{2}}$, then $\frac{du}{dx}=\frac{1}{2}(2x + 7)^{-\frac{1}{2}}\times2=(2x + 7)^{-\frac{1}{2}}$.

Step4: Apply the chain - rule

The chain - rule states that $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substituting $\frac{dy}{du}=e^{u}$ and $\frac{du}{dx}=(2x + 7)^{-\frac{1}{2}}$ and replacing $u=\sqrt{2x + 7}$, we get $\frac{dy}{dx}=e^{\sqrt{2x + 7}}\cdot\frac{1}{\sqrt{2x + 7}}$.

Answer:

$\frac{e^{\sqrt{2x + 7}}}{\sqrt{2x + 7}}$