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

To find the derivative of the function \( f(x) = 4x \) from first principles, we need to use the definition of the derivative:

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

### Step 1: Set up the difference quotient
We know \( f(x) = 4x \), so let's calculate \( f(x+h) \):

\[
f(x+h) = 4(x+h) = 4x + 4h
\]

Now, substitute into the difference quotient:

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

### Step 2: Simplify the expression
Simplify the numerator:

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

Cancel \( h \) from the numerator and denominator:

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

### Step 3: Evaluate the limit
Since there are no \( h \)-terms left, the limit is simply:

\[
f'(x) = 4
\]

### Intuition behind the result
The function \( f(x) = 4x \) is a linear function with a constant slope of 4. The derivative represents the rate of change of \( f(x) \) with respect to \( x \). 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 \( x \), \( f(x) \) 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 \( f(x) = 4x \)?
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 \( f(x) = mx + b \), the derivative is always equal to the slope \( m \).

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

Explore how to find the derivative of the linear function f(x) = 4x using the first principles of calculus. This guide explains each step, from setting up the difference quotient to understanding the intuition behind the result. Suitable for Grades 11-12 and early university students learning calculus.

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