Math Problem Statement

given the function f(x) = 4x. find the derivative form first principles, and explain why your answer makes sense intuitively.

Solution

To find the derivative of the function f(x)=4xf(x) = 4x from first principles, we need to 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: Set up the difference quotient

We know f(x)=4xf(x) = 4x, so let's calculate f(x+h)f(x+h):

f(x+h)=4(x+h)=4x+4hf(x+h) = 4(x+h) = 4x + 4h

Now, substitute into the difference quotient:

f(x)=limh0(4x+4h)4xhf'(x) = \lim_{h \to 0} \frac{(4x + 4h) - 4x}{h}

Step 2: Simplify the expression

Simplify the numerator:

f(x)=limh04x+4h4xh=limh04hhf'(x) = \lim_{h \to 0} \frac{4x + 4h - 4x}{h} = \lim_{h \to 0} \frac{4h}{h}

Cancel hh from the numerator and denominator:

f(x)=limh04f'(x) = \lim_{h \to 0} 4

Step 3: Evaluate the limit

Since there are no hh-terms left, the limit is simply:

f(x)=4f'(x) = 4

Intuition behind the result

The function f(x)=4xf(x) = 4x is a linear function with a constant slope of 4. The derivative represents the rate of change of f(x)f(x) with respect to xx. Since the function is linear, the rate of change is constant, and this is reflected by the derivative being equal to 4.

This makes sense intuitively because for every unit increase in xx, f(x)f(x) increases by 4, which aligns with the definition of the derivative as the slope of the function.

Would you like more details on any part of the explanation?

Related Questions:

  1. How does the definition of the derivative apply to non-linear functions?
  2. What is the geometric interpretation of the derivative for a linear function like f(x)=4xf(x) = 4x?
  3. How do you differentiate a quadratic function from first principles?
  4. What are some common mistakes when using the first principles approach to finding a derivative?
  5. How does the derivative of a constant function differ from that of a linear function?

Tip:

Always remember that for linear functions like f(x)=mx+bf(x) = mx + b, the derivative is always equal to the slope mm.

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Math Problem Analysis

Mathematical Concepts

Calculus
Derivatives
First Principles of Derivatives

Formulas

f'(x) = lim(h -> 0) [(f(x+h) - f(x)) / h]

Theorems

Definition of the Derivative

Suitable Grade Level

Grades 11-12, Early University