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To find \(\frac{dy}{dx}\) for the given function:

\[
y = \frac{(5x^2 + 1)^2}{5\sqrt{x}}
\]

we can proceed by using the quotient rule and chain rule.

### Step 1: Rewrite the equation
First, rewrite the square root of \(x\) as a power:

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

### Step 2: Apply the Quotient Rule
The quotient rule states:

\[
\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}
\]

where \(u = (5x^2 + 1)^2\) and \(v = 5x^{1/2}\).

### Step 3: Differentiate \(u\) (numerator)
To differentiate \(u = (5x^2 + 1)^2\), use the chain rule. Let \(z = 5x^2 + 1\), so \(u = z^2\).

\[
\frac{du}{dx} = 2z \cdot \frac{dz}{dx} = 2(5x^2 + 1) \cdot (10x) = 20x(5x^2 + 1)
\]

### Step 4: Differentiate \(v\) (denominator)
To differentiate \(v = 5x^{1/2}\), apply the power rule:

\[
\frac{dv}{dx} = 5 \cdot \frac{1}{2}x^{-1/2} = \frac{5}{2}x^{-1/2}
\]

### Step 5: Apply the Quotient Rule
Now, apply the quotient rule:

\[
\frac{dy}{dx} = \frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2}
\]

Substitute the values of \(\frac{du}{dx}\), \(\frac{dv}{dx}\), \(u\), and \(v\):

\[
\frac{dy}{dx} = \frac{5x^{1/2} \cdot 20x(5x^2 + 1) - (5x^2 + 1)^2 \cdot \frac{5}{2}x^{-1/2}}{(5x^{1/2})^2}
\]

### Step 6: Simplify
We can simplify the expression, but for now, this is the general form of the derivative.

Would you like me to continue simplifying this expression further or clarify any part of the steps?

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Here are some questions to expand on this:

1. How would the derivative change if the denominator were a different function?
2. Can you apply the product rule for cases where both terms are products?
3. How would you apply the chain rule if there were more nested functions in the numerator?
4. How does the power rule help simplify roots when differentiating?
5. Can you apply this technique to other rational functions with powers?

**Tip:** When differentiating complex functions, it's helpful to break them into smaller parts (e.g., quotient rule or chain rule) to simplify the process.

Learn how to find the derivative of y = (5x^2 + 1)^2 / 5√x using the quotient rule, chain rule, and power rule. This detailed step-by-step solution covers key differentiation techniques suitable for students in Grades 11-12 or early college.

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