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To solve for the derivative \( \frac{dy}{dx} \) of the given function:

\[
y = \sqrt{x(x^5 + 5)}
\]

### Step 1: Simplify the function
First, rewrite the square root as a power of \( \frac{1}{2} \):

\[
y = \left[ x(x^5 + 5) \right]^{1/2}
\]

### Step 2: Apply the product rule and chain rule
Now, we have a product inside the square root. Let’s denote the function inside the square root as:

\[
u(x) = x(x^5 + 5)
\]

The derivative of \( y \) can be found using the chain rule:

\[
\frac{dy}{dx} = \frac{1}{2} \left[ x(x^5 + 5) \right]^{-1/2} \cdot \frac{d}{dx} \left[ x(x^5 + 5) \right]
\]

### Step 3: Differentiate the product inside
Next, differentiate the product \( u(x) = x(x^5 + 5) \) using the product rule:

\[
\frac{d}{dx} \left[ x(x^5 + 5) \right] = (x^5 + 5) \cdot \frac{d}{dx}(x) + x \cdot \frac{d}{dx}(x^5 + 5)
\]

The derivatives are as follows:
- \( \frac{d}{dx}(x) = 1 \)
- \( \frac{d}{dx}(x^5 + 5) = 5x^4 \)

Thus, we get:

\[
\frac{d}{dx} \left[ x(x^5 + 5) \right] = (x^5 + 5) + x(5x^4) = x^5 + 5 + 5x^5 = 6x^5 + 5
\]

### Step 4: Substitute back into the chain rule
Now substitute this derivative back into the chain rule expression:

\[
\frac{dy}{dx} = \frac{1}{2} \left[ x(x^5 + 5) \right]^{-1/2} \cdot (6x^5 + 5)
\]

### Final Answer:

\[
\frac{dy}{dx} = \frac{6x^5 + 5}{2 \sqrt{x(x^5 + 5)}}
\]

Let me know if you'd like more clarification or a detailed explanation on any step!

Here are five related questions you might consider:
1. How does the product rule work in this context?
2. What role does the chain rule play in finding this derivative?
3. Can we apply logarithmic differentiation for a similar problem?
4. How would the derivative change if we had a cube root instead of a square root?
5. What if the function inside the square root had an additional constant multiplier?

**Tip:** Always simplify your function before differentiating to make the process smoother and avoid errors.

Find the derivative of y = √x(x^5 + 5).

Learn how to find the derivative of y = √x(x^5 + 5) using the product and chain rules. This calculus problem is suitable for Grades 11-12 students and covers key concepts such as differentiation, product rule, and chain rule.

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