To find the derivative $f'(x)$ of the function $f(x) = -x^2 + 5x - 9$ using the definition of the derivative, we need to compute the following limit:

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

Step 1: Compute $f(x + h)$

Substitute $x + h$ into the function $f(x)$:

$f(x + h) = -(x + h)^2 + 5(x + h) - 9$ Expand this expression: $f(x + h) = -\left( x^2 + 2xh + h^2 \right) + 5x + 5h - 9$ Simplifying: $f(x + h) = -x^2 - 2xh - h^2 + 5x + 5h - 9$

Step 2: Compute $f(x + h) - f(x)$

Now subtract $f(x) = -x^2 + 5x - 9$ from $f(x + h)$: $f(x + h) - f(x) = \left( -x^2 - 2xh - h^2 + 5x + 5h - 9 \right) - \left( -x^2 + 5x - 9 \right)$ Distribute the negative sign: $f(x + h) - f(x) = -x^2 - 2xh - h^2 + 5x + 5h - 9 + x^2 - 5x + 9$ Simplify the terms: $f(x + h) - f(x) = -2xh - h^2 + 5h$

Step 3: Form the difference quotient

Now, divide by $h$: $\frac{f(x + h) - f(x)}{h} = \frac{-2xh - h^2 + 5h}{h}$ Simplify the expression: $\frac{f(x + h) - f(x)}{h} = -2x - h + 5$

Step 4: Take the limit as $h \to 0$

Now take the limit as $h$ approaches 0: $f'(x) = \lim_{{h \to 0}} (-2x - h + 5) = -2x + 5$

Thus, the derivative of $f(x)$ is: $f'(x) = -2x + 5$

Step 5: Find $f'(1)$, $f'(2)$, and $f'(3)$

• $f'(1) = -2(1) + 5 = -2 + 5 = 3$
• $f'(2) = -2(2) + 5 = -4 + 5 = 1$
• $f'(3) = -2(3) + 5 = -6 + 5 = -1$

Summary:

• $f'(x) = -2x + 5$
• $f'(1) = 3$
• $f'(2) = 1$
• $f'(3) = -1$

Would you like more details on any of these steps or further explanations?

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Related Questions:

1. What is the geometric meaning of the derivative $f'(x)$ at a point?
2. How would the second derivative of $f(x) = -x^2 + 5x - 9$ be computed?
3. What is the interpretation of the values $f'(1)$, $f'(2)$, and $f'(3)$?
4. Can we use the power rule to differentiate $f(x) = -x^2 + 5x - 9$ and get the same result?
5. How would the derivative change if the function were $f(x) = -x^3 + 5x - 9$?

Tip: The definition of the derivative is a fundamental concept in calculus that captures the rate of change of a function at any point. It’s crucial for understanding motion, optimization, and various other mathematical applications.