Let's analyze the given problem.

We are asked to find the derivative of the function:

$f(x) = 4 \sqrt{5x^2 + 3}.$

Step 1: Rewrite the square root in exponential form

We know that:

$\sqrt{u} = u^{1/2}.$

So, the function becomes:

$f(x) = 4 \cdot (5x^2 + 3)^{1/2}.$

Step 2: Use the Chain Rule

The chain rule states that:

$\frac{d}{dx} \left[ f(g(x)) \right] = f'(g(x)) \cdot g'(x).$

Here, $f(u) = u^{1/2}$ with $u = 5x^2 + 3$. Let’s differentiate step by step.

Step 3: Differentiate the outer function $(5x^2 + 3)^{1/2}$

Using the power rule:

$\frac{d}{du} \left( u^{1/2} \right) = \frac{1}{2} u^{-1/2}.$

So:

$\frac{d}{dx} \left[ (5x^2 + 3)^{1/2} \right] = \frac{1}{2} (5x^2 + 3)^{-1/2} \cdot \frac{d}{dx}(5x^2 + 3).$

Step 4: Differentiate the inner function $5x^2 + 3$

$\frac{d}{dx}(5x^2 + 3) = 10x.$

Step 5: Combine the results

Now, multiply everything together:

$\frac{d}{dx} \left[ (5x^2 + 3)^{1/2} \right] = \frac{1}{2} (5x^2 + 3)^{-1/2} \cdot 10x.$

This simplifies to:

$\frac{5x}{(5x^2 + 3)^{1/2}}.$

Step 6: Apply the constant multiplier

Since the original function is $f(x) = 4 \cdot (5x^2 + 3)^{1/2}$, we multiply the result by 4:

$f'(x) = 4 \cdot \frac{5x}{(5x^2 + 3)^{1/2}}.$

This simplifies to:

$f'(x) = \frac{20x}{\sqrt{5x^2 + 3}}.$

Final Answer

The derivative of the function $f(x) = 4 \sqrt{5x^2 + 3}$ is:

$f'(x) = \frac{20x}{\sqrt{5x^2 + 3}}.$

Would you like me to walk through any step in more detail?

Here are 5 related questions to deepen your understanding:

1. How does the chain rule work for composite functions with more than two layers?
2. What is the derivative of $\sqrt{3x^2 - 2x + 1}$?
3. Can you apply the product rule and chain rule together in one example?
4. What is the second derivative of the given function?
5. How do you handle derivatives involving cube roots?

Tip: When applying the chain rule, always differentiate the outer function first, leaving the inner function intact, and then multiply by the derivative of the inner function.