To find the derivative of the function:

$y = \sqrt{6 + 6e^{7x}}$

We proceed as follows:

Step 1: Rewrite the function

Rewrite the square root using the power rule: $y = \left(6 + 6e^{7x}\right)^{\frac{1}{2}}$

Step 2: Differentiate using the chain rule

Let $u = 6 + 6e^{7x}$. Then $y = u^{\frac{1}{2}}$, so:

$\frac{dy}{dx} = \frac{1}{2}u^{-\frac{1}{2}} \cdot \frac{du}{dx}$

Now compute $\frac{du}{dx}$.

Step 3: Differentiate $u = 6 + 6e^{7x}$

The derivative of $u$ is: $\frac{du}{dx} = 6 \cdot \frac{d}{dx}\left(e^{7x}\right)$

The derivative of $e^{7x}$ is $7e^{7x}$, so: $\frac{du}{dx} = 6 \cdot 7e^{7x} = 42e^{7x}$

Step 4: Substitute back

Substitute $\frac{du}{dx}$ and $u$ into $\frac{dy}{dx}$: $\frac{dy}{dx} = \frac{1}{2} \left(6 + 6e^{7x}\right)^{-\frac{1}{2}} \cdot 42e^{7x}$

Step 5: Simplify

$\frac{dy}{dx} = \frac{42e^{7x}}{2\sqrt{6 + 6e^{7x}}}$

$\frac{dy}{dx} = \frac{21e^{7x}}{\sqrt{6 + 6e^{7x}}}$

Final Answer:

$\boxed{\frac{dy}{dx} = \frac{21e^{7x}}{\sqrt{6 + 6e^{7x}}}}$

Let me know if you'd like further explanation or assistance with related questions!

Here are 5 related questions to explore:

1. How would you find the second derivative of this function?
2. What happens to the derivative as $x \to \infty$?
3. Can you apply implicit differentiation to solve this problem differently?
4. How does this derivative compare to simpler cases like $y = \sqrt{e^x}$?
5. How would you compute the derivative if the base function were a cubic root instead?

Tip: When differentiating composite functions, always remember to apply the chain rule systematically!