Let's break down and address each of the two questions based on the image:

1. Derivative using the difference quotient

The first question asks which expression represents $f'(x)$ for the function:

$f(x) = 3x^2 - 5\sqrt{x}$

The derivative $f'(x)$ is defined as the limit:

$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

Given the function $f(x) = 3x^2 - 5\sqrt{x}$, we calculate $f(x+h)$:

$f(x+h) = 3(x+h)^2 - 5\sqrt{x+h}$

Substitute this expression into the difference quotient:

$f'(x) = \lim_{h \to 0} \frac{(3(x+h)^2 - 5\sqrt{x+h}) - (3x^2 - 5\sqrt{x})}{h}$

The correct expression matching this form from the options provided in the image is:

$\boxed{\lim_{h \to 0} \frac{(3(x+h)^2 - 5\sqrt{x+h}) - (3x^2 - 5\sqrt{x})}{h}}$

Thus, the correct answer for this part is the first option.

2. Difference quotient for specific values

The second question involves the function $f(x) = 2x^2 - 3x$ and asks for $f'(c)$ where $c = 2$. To do this, we must use the definition of the derivative:

$f'(c) = \lim_{h \to 0} \frac{f(c+h) - f(c)}{h}$

Given $c = 2$, the function becomes $f(x) = 2x^2 - 3x$. We first calculate $f(2)$:

$f(2) = 2(2)^2 - 3(2) = 8 - 6 = 2$

Next, we calculate $f(2+h)$:

$f(2+h) = 2(2+h)^2 - 3(2+h)$ Expanding this:

$f(2+h) = 2(4 + 4h + h^2) - 3(2+h)$ $f(2+h) = 8 + 8h + 2h^2 - 6 - 3h$ $f(2+h) = 2 + 5h + 2h^2$

Now, use the difference quotient:

$f'(2) = \lim_{h \to 0} \frac{(2 + 5h + 2h^2) - 2}{h} = \lim_{h \to 0} \frac{5h + 2h^2}{h}$ Simplify:

$f'(2) = \lim_{h \to 0} (5 + 2h) = 5$

Thus, the answer for $f'(2)$ is $\boxed{5}$.

Do you have any questions or would you like further details?

Related Questions:

1. How would you apply the difference quotient for a trigonometric function like $\sin(x)$?
2. Can you find the second derivative $f''(x)$ for the function $f(x) = 3x^2 - 5\sqrt{x}$?
3. What happens if you calculate the difference quotient for a polynomial function like $x^3$?
4. How does the chain rule relate to the derivative of $5\sqrt{x}$?
5. Can you generalize the difference quotient for functions of multiple variables?

Tip:

Always expand terms carefully when working with the difference quotient to ensure correct simplifications for limits.