On a piece of paper, determine from the first principles,

f′(x)   if   f(x)=−2x.
Then based on your working, what is the correct expression for

 
f(x+h)−f(x)?

To determine the derivative of \( f(x) = -2x \) from first principles, we use the definition of the derivative:

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

### Step 1: Compute \( f(x+h) \)
Given \( f(x) = -2x \), we substitute \( x+h \) into the function:

\[
f(x+h) = -2(x+h) = -2x - 2h
\]

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

\[
f(x+h) - f(x) = (-2x - 2h) - (-2x)
\]

Simplifying this:

\[
f(x+h) - f(x) = -2h
\]

### Step 3: Derive the expression for \( f'(x) \)
Substitute the expression \( -2h \) into the limit definition:

\[
f'(x) = \lim_{h \to 0} \frac{-2h}{h}
\]

Simplifying:

\[
f'(x) = \lim_{h \to 0} -2 = -2
\]

So, the derivative of \( f(x) = -2x \) is \( f'(x) = -2 \).

### Answer to the Question:
Based on the working, the correct expression for \( f(x+h) - f(x) \) is:

\[
f(x+h) - f(x) = -2h
\]

Would you like to go over any details or have further questions?

Here are 5 related questions you might find useful:
1. What is the geometric interpretation of the derivative \( f'(x) \)?
2. How does the result change if \( f(x) = 3x + 5 \)?
3. Can you find the derivative of \( f(x) = x^2 \) from first principles?
4. How would the process differ if the function involved a higher degree polynomial?
5. What are the implications of a constant derivative value for \( f(x) = -2x \)?

**Tip:** When applying the first principles of calculus, always simplify the expression \( f(x+h) - f(x) \) as much as possible before taking the limit.

Learn how to calculate the derivative of f(x) = -2x using first principles. Understand the definition of derivatives and how to apply it step-by-step. Suitable for advanced high school students and calculus enthusiasts.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation