given f(x)=-3x+10 complete the following steps to find f'(x)=lim (x-->0) f(x+h)-f(x)/h

To find \( f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \), we need to go through the following steps:

### Step 1: Find \( f(x+h) \)
We are given the function \( f(x) = -3x + 10 \). Now, substitute \( x+h \) into the function:

\[
f(x+h) = -3(x+h) + 10 = -3x - 3h + 10
\]

### Step 2: Compute \( f(x+h) - f(x) \)
Now, subtract \( f(x) \) from \( f(x+h) \):

\[
f(x+h) - f(x) = (-3x - 3h + 10) - (-3x + 10)
\]

Simplify the expression:

\[
f(x+h) - f(x) = -3x - 3h + 10 + 3x - 10 = -3h
\]

### Step 3: Set up the difference quotient
Now, divide \( f(x+h) - f(x) \) by \( h \):

\[
\frac{f(x+h) - f(x)}{h} = \frac{-3h}{h}
\]

Simplify:

\[
\frac{f(x+h) - f(x)}{h} = -3
\]

### Step 4: Take the limit as \( h \to 0 \)
The next step is to take the limit of the difference quotient as \( h \to 0 \):

\[
f'(x) = \lim_{h \to 0} -3 = -3
\]

### Final Result:
The derivative of the function is constant:

\[
f'(x) = -3
\]

Would you like further details or clarifications?

Here are 5 related questions for further practice:
1. What is the geometric interpretation of the derivative \( f'(x) \)?
2. How would the derivative change if the function were \( f(x) = -3x + c \) with a different constant \( c \)?
3. What is the second derivative \( f''(x) \) of \( f(x) = -3x + 10 \)?
4. How does the derivative help in understanding the behavior of linear functions?
5. What happens to the slope when you modify the coefficient of \( x \) in a linear function?

**Tip**: For linear functions of the form \( f(x) = mx + b \), the derivative \( f'(x) \) is always equal to the slope \( m \).

Explore how to find the derivative of the linear function f(x) = -3x + 10 using the limit definition. This step-by-step solution covers key calculus concepts such as limits and the derivative, suitable for students in Grades 11-12.

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