Math Problem Statement

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)?

Solution

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

f(x)=limh0f(x+h)f(x)hf'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

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

Given f(x)=2xf(x) = -2x, we substitute x+hx+h into the function:

f(x+h)=2(x+h)=2x2hf(x+h) = -2(x+h) = -2x - 2h

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

Now, subtract f(x)f(x) from f(x+h)f(x+h):

f(x+h)f(x)=(2x2h)(2x)f(x+h) - f(x) = (-2x - 2h) - (-2x)

Simplifying this:

f(x+h)f(x)=2hf(x+h) - f(x) = -2h

Step 3: Derive the expression for f(x)f'(x)

Substitute the expression 2h-2h into the limit definition:

f(x)=limh02hhf'(x) = \lim_{h \to 0} \frac{-2h}{h}

Simplifying:

f(x)=limh02=2f'(x) = \lim_{h \to 0} -2 = -2

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

Answer to the Question:

Based on the working, the correct expression for f(x+h)f(x)f(x+h) - f(x) is:

f(x+h)f(x)=2hf(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)f'(x)?
  2. How does the result change if f(x)=3x+5f(x) = 3x + 5?
  3. Can you find the derivative of f(x)=x2f(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)=2xf(x) = -2x?

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

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Math Problem Analysis

Mathematical Concepts

Calculus
Derivatives
Limits

Formulas

Definition of derivative

Theorems

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Suitable Grade Level

Grades 11-12