Let's break this down carefully and go through the steps of finding the first and second derivatives of the function.

The given function is:

$f(x) = \frac{x^2}{7 + 4x}$

1. First Derivative $f'(x)$

We'll use the quotient rule for derivatives to differentiate $f(x)$. The quotient rule states that:

$\frac{d}{dx} \left( \frac{u(x)}{v(x)} \right) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}$

Here, $u(x) = x^2$ and $v(x) = 7 + 4x$.

First, calculate the derivatives of $u(x)$ and $v(x)$:

$u'(x) = \frac{d}{dx}(x^2) = 2x$ $v'(x) = \frac{d}{dx}(7 + 4x) = 4$

Now, apply the quotient rule:

$f'(x) = \frac{(2x)(7 + 4x) - (x^2)(4)}{(7 + 4x)^2}$

Simplify the numerator:

$f'(x) = \frac{2x(7 + 4x) - 4x^2}{(7 + 4x)^2}$ $f'(x) = \frac{14x + 8x^2 - 4x^2}{(7 + 4x)^2}$ $f'(x) = \frac{14x + 4x^2}{(7 + 4x)^2}$

So the first derivative is:

$f'(x) = \frac{4x^2 + 14x}{(7 + 4x)^2}$

2. Second Derivative $f''(x)$

Now, we differentiate $f'(x)$ again to find the second derivative. We'll use the quotient rule again, where:

$u(x) = 4x^2 + 14x$ $v(x) = (7 + 4x)^2$

First, calculate the derivatives of $u(x)$ and $v(x)$:

$u'(x) = \frac{d}{dx}(4x^2 + 14x) = 8x + 14$ $v'(x) = \frac{d}{dx}[(7 + 4x)^2] = 2(7 + 4x)(4) = 8(7 + 4x)$

Now, apply the quotient rule:

$f''(x) = \frac{(8x + 14)(7 + 4x)^2 - (4x^2 + 14x)(8(7 + 4x))}{(7 + 4x)^4}$

Simplify the expression for $f''(x)$. The final result after simplification will look like:

$f''(x) = \frac{(7 + 4x)(8x + 14) - (4x^2 + 14x)(8)}{(7 + 4x)^3}$

Thus, the second derivative is:

$f''(x) = \frac{(7 + 4x)(8x + 14) - (4x^2 + 14x)(8)}{(7 + 4x)^3}$

Summary:

• First derivative: $f'(x) = \frac{4x^2 + 14x}{(7 + 4x)^2}$
• Second derivative: $f''(x) = \frac{(7 + 4x)(8x + 14) - (4x^2 + 14x)(8)}{(7 + 4x)^3}$

Would you like to go over any specific steps in more detail or clarify any part of the solution?

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Here are five related questions you can explore:

1. How does the quotient rule differ from the product rule in differentiation?
2. What are some other functions where the quotient rule would apply?
3. Can we simplify the second derivative further in this case?
4. What is the importance of higher-order derivatives in calculus?
5. How would this function behave as $x \to \infty$?

Tip: Always check for opportunities to simplify the numerator when applying the quotient rule to avoid complex expressions!