help me find the derivative of f(x)=7-3x-x^2 using lim h->0 (7-3x-x^2+h)-(7-3x-x^2)/h

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.

Learn how to find the derivative of the function f(x) = 7 - 3x - x^2 using the limit definition. This step-by-step solution covers essential calculus concepts, including limits and algebraic simplification, suitable for high school or introductory college-level students.

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