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

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

Step 1: Find 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).f(x+h) = -2(x+h).

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

Now, compute f(x+h)f(x)f(x+h) - f(x):

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

Simplify the expression:

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

The 2x-2x and +2x+2x cancel each other out:

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

So, the expression for f(x+h)f(x)f(x+h) - f(x) is:

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

Step 3: Compute the Derivative Using the Limit Definition

Substitute the expression we found into the limit definition of the derivative:

f(x)=limh02hh.f'(x) = \lim_{h \to 0} \frac{-2h}{h}.

Simplify the fraction:

f(x)=limh0(2)=2.f'(x) = \lim_{h \to 0} (-2) = -2.

Conclusion

The derivative of f(x)=2xf(x) = -2x is f(x)=2f'(x) = -2.

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

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

Would you like more details on any step?

Here are 5 related questions you might consider:

  1. How do you apply the first principle to find the derivative of more complex functions?
  2. Why does the expression for f(x)f'(x) simplify to just a constant in this case?
  3. How can we interpret the derivative f(x)=2f'(x) = -2 geometrically?
  4. What happens to the derivative if the function is f(x)=2x2f(x) = -2x^2?
  5. How does this method of finding derivatives compare with using differentiation rules?

Tip: The derivative represents the rate of change of a function; understanding its geometric interpretation can enhance comprehension.

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Math Problem Analysis

Mathematical Concepts

Calculus
Derivatives
Limit Definition

Formulas

Definition of the derivative

Theorems

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Suitable Grade Level

Advanced High School