Differentiate the function.
y = 

SQUre root of 6 + 6e6x
 
y' = 


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.

Differentiate the function. y = Square root of 6 + 6e^6x

Learn how to differentiate the function y = √(6 + 6e^6x) step by step using the chain rule. This problem covers key concepts in differentiation, including applying the chain rule and handling exponential functions. Suitable for students in Grades 11-12 or early calculus learners.

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