To differentiate the function:

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

we apply the chain rule. Here's the step-by-step process:

Step 1: Express the function in a more convenient form

$y = (6 + 6e^{6x})^{\frac{1}{2}}$

Step 2: Apply the chain rule

The derivative of $y = u^{\frac{1}{2}}$ with respect to $u$ is:

$\frac{dy}{du} = \frac{1}{2}u^{-\frac{1}{2}} = \frac{1}{2}(6 + 6e^{6x})^{-\frac{1}{2}}$

Now, we need to multiply by the derivative of the inside expression $6 + 6e^{6x}$.

Step 3: Differentiate the inside expression

The derivative of $6 + 6e^{6x}$ with respect to $x$ is:

[ \frac{d}{dx}[6 + 6e^{6x}] = 6 \cdot \frac{d}{dx}[e^{6x}] = 6 \cdot 6e^{6x} = 36e^{6x} ]

Step 4: Multiply both parts

Now, combine the derivative of the outer and inner functions:

$y' = \frac{1}{2}(6 + 6e^{6x})^{-\frac{1}{2}} \cdot 36e^{6x}$

Step 5: Simplify the expression

$y' = \frac{36e^{6x}}{2(6 + 6e^{6x})^{\frac{1}{2}}}$ $y' = \frac{18e^{6x}}{\sqrt{6 + 6e^{6x}}}$

Thus, the derivative of the function is:

$y' = \frac{18e^{6x}}{\sqrt{6 + 6e^{6x}}}$

Would you like more details, or do you have any questions?

Here are 5 related questions for further practice:

1. Differentiate $y = \sqrt{3 + e^{3x}}$.
2. Differentiate $y = \sqrt{7x + e^{7x}}$.
3. What is the derivative of $y = e^{3x}\sqrt{5x + 2}$?
4. Differentiate $y = \ln(6 + e^{6x})$.
5. What is the second derivative of $y = \sqrt{6 + 6e^{6x}}$?

Tip: Always simplify your expressions as much as possible after differentiating to make the result more manageable.