The image you uploaded covers an example of using the Quotient Rule for derivatives. The quotient rule is a method to differentiate functions that are written as the ratio of two other functions.

The example provided shows the function:

$f(x) = \frac{6x^3}{2x + 5}$

Quotient Rule Formula

If you have a function $f(x) = \frac{g(x)}{h(x)}$, the derivative is given by:

$f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2}$

Where:

• $g(x)$ is the numerator function.
• $h(x)$ is the denominator function.
• $g'(x)$ is the derivative of the numerator.
• $h'(x)$ is the derivative of the denominator.

Applying the Quotient Rule

For the given function $f(x) = \frac{6x^3}{2x + 5}$:

1. Identify $g(x)$ and $h(x)$:

   • $g(x) = 6x^3$
   • $h(x) = 2x + 5$

2. Find the derivatives:

   • $g'(x) = 18x^2$ (since $\frac{d}{dx}[6x^3] = 18x^2$)
   • $h'(x) = 2$ (since $\frac{d}{dx}[2x + 5] = 2$)

3. Apply the Quotient Rule formula: $f'(x) = \frac{(18x^2)(2x + 5) - (6x^3)(2)}{(2x + 5)^2}$

4. Simplify:

   • First, expand the terms in the numerator: $f'(x) = \frac{18x^2(2x + 5) - 12x^3}{(2x + 5)^2}$
   • Now, distribute $18x^2$ in the first term: $f'(x) = \frac{(36x^3 + 90x^2) - 12x^3}{(2x + 5)^2}$
   • Combine like terms: $f'(x) = \frac{24x^3 + 90x^2}{(2x + 5)^2}$

So the derivative of the function is:

$f'(x) = \frac{24x^3 + 90x^2}{(2x + 5)^2}$

Would you like further details or explanations on any part of this solution?

Here are 5 related questions you could explore:

1. How does the quotient rule differ from the product rule?
2. Can the quotient rule be applied to functions with trigonometric terms?
3. What happens when the denominator function is constant in the quotient rule?
4. How can we verify if the derivative result is correct?
5. What are other methods to simplify or manipulate functions before applying the quotient rule?

Tip: Always check if the quotient can be simplified before using the quotient rule—it might save you some work!