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.