https://file.aiquickdraw.com/mathbot/file/1726107524333qwie641j.jpg

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.

Given f(x) = 3x^2 - 5√x, which of the following expressions will represent f'(x)? Also, given f(x) = 2x^2 - 3x and c = 2, find f'(c) using the difference quotient and taking the limit as h approaches 0.

Learn how to find the derivative of functions such as f(x) = 3x^2 - 5√x using the difference quotient. This step-by-step explanation involves applying the limit as h approaches 0 to calculate f'(x) and f'(2).

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