What is the derivative of (x) ?

The derivative of a function measures how that function changes as its input changes. For the function \( f(x) = x \), the derivative is quite straightforward.

Calculating the Derivative

To find the derivative of \( f(x) = x \), we can use the basic rules of differentiation. The derivative of \( x \) with respect to \( x \) is:

• f'(x) = 1

Why is this the case?

This result means that for every unit increase in \( x \), the value of the function \( f(x) \) increases by 1. In simpler terms, the slope of the line represented by \( f(x) = x \) is constant and equal to 1.

Graphical Representation

If you were to graph the function \( f(x) = x \), it would be a straight line that passes through the origin (0,0) with a slope of 1. This visual can help reinforce the concept of the derivative being the slope of the function at any point.

In summary, the derivative of \( x \) is simply 1, indicating a constant rate of change. This is a fundamental concept in calculus that lays the groundwork for more complex functions.