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We are given a function \( f(x) = 3 - 5x^2 \), and the task is to:

1. Simplify the difference quotient \( \frac{f(3 + h) - f(3)}{h} \).
2. Then, determine the derivative \( f'(3) \) by calculating the limit:

\[
\lim_{h \to 0} \frac{f(3 + h) - f(3)}{h}.
\]

### Step 1: Simplify \( f(3 + h) - f(3) \)

First, we need to calculate \( f(3 + h) \) and \( f(3) \).

- \( f(x) = 3 - 5x^2 \)
  
So,

\[
f(3 + h) = 3 - 5(3 + h)^2 = 3 - 5(9 + 6h + h^2) = 3 - 5(9) - 5(6h) - 5(h^2)
\]
\[
f(3 + h) = 3 - 45 - 30h - 5h^2 = -42 - 30h - 5h^2
\]

Now calculate \( f(3) \):

\[
f(3) = 3 - 5(3^2) = 3 - 5(9) = 3 - 45 = -42
\]

### Step 2: Simplify the difference

Now we compute the difference \( f(3 + h) - f(3) \):

\[
f(3 + h) - f(3) = (-42 - 30h - 5h^2) - (-42) = -30h - 5h^2
\]

Thus, the difference quotient becomes:

\[
\frac{f(3 + h) - f(3)}{h} = \frac{-30h - 5h^2}{h} = -30 - 5h
\]

### Step 3: Find the limit as \( h \to 0 \)

Now, take the limit of the expression \( -30 - 5h \) as \( h \to 0 \):

\[
\lim_{h \to 0} (-30 - 5h) = -30
\]

### Conclusion:

The derivative \( f'(3) = -30 \).

Would you like further clarification or details on any part of this process?

Here are 5 related questions to deepen your understanding:
1. How would you calculate \( f'(x) \) for any general value of \( x \)?
2. What is the geometric interpretation of \( f'(3) \) in this context?
3. How does the quadratic term \( -5x^2 \) affect the slope of the function?
4. How can you apply the difference quotient method to find derivatives of more complex functions?
5. How does the limit process help us define instantaneous rates of change?

**Tip**: The difference quotient is foundational in understanding derivatives, which represent the slope of the tangent line at any point on a curve.

Soit une fonction f définie par : f(x) = 3 - 5x^2. Calculer et simplifier l'expression (f(3 + h) - f(3)) / h. En calculant lim (h→0) (f(3 + h) - f(3)) / h, déterminer f'(3).

Learn how to calculate the derivative of the quadratic function f(x) = 3 - 5x^2 at x = 3. This problem covers key calculus concepts including limits, difference quotients, and the definition of the derivative.

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