To find the derivative of the function $f(x) = 7 - 3x - x^2$ using the limit definition of a derivative, we'll follow the process step by step.

The limit definition of a derivative is given by: $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

You already have the function set up: $\frac{(7 - 3(x+h) - (x+h)^2) - (7 - 3x - x^2)}{h}$

Now, let's work through it step by step:

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

$f(x+h) = 7 - 3(x+h) - (x+h)^2$ Expand both terms: $f(x+h) = 7 - 3x - 3h - (x^2 + 2xh + h^2)$ $f(x+h) = 7 - 3x - 3h - x^2 - 2xh - h^2$

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

Now subtract $f(x) = 7 - 3x - x^2$ from $f(x+h)$: $f(x+h) - f(x) = (7 - 3x - 3h - x^2 - 2xh - h^2) - (7 - 3x - x^2)$ Distribute the negative sign: $f(x+h) - f(x) = 7 - 3x - 3h - x^2 - 2xh - h^2 - 7 + 3x + x^2$ Simplify the terms: $f(x+h) - f(x) = -3h - 2xh - h^2$

Step 3: Divide by $h$

Now divide the expression by $h$: $\frac{f(x+h) - f(x)}{h} = \frac{-3h - 2xh - h^2}{h}$ Simplify each term: $\frac{f(x+h) - f(x)}{h} = -3 - 2x - h$

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

Now, take the limit of the expression as $h \to 0$: $\lim_{h \to 0} (-3 - 2x - h) = -3 - 2x$

Thus, the derivative of $f(x) = 7 - 3x - x^2$ is: $f'(x) = -3 - 2x$

Would you like more details or have any further questions?

Here are 5 related questions:

1. How do we apply the product rule in derivatives?
2. What is the derivative of a sum of functions?
3. Can you explain how to differentiate $f(x) = x^3 - 5x^2 + 6x - 2$?
4. What is the meaning of the second derivative of a function?
5. How do you find the derivative of a trigonometric function?

Tip: When finding derivatives using the limit definition, always simplify the expression as much as possible before applying the limit.