The derivative shows us how a function is changing at a specific spot.

What It Means:

• In simple terms, the derivative ( f'(x) ) is defined as:

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

(But we don't need to worry too much about the math symbols right now!)

How to Understand It:

1. Slope of the Tangent Line: The derivative tells us the steepness of a line that just touches the curve at the point ( (x, f(x)) ).
2. Speed of Change: It shows us how quickly the function is changing at that specific point.

An Example:

If we take ( f(x) = x^2 ), the derivative ( f'(x) = 2x ) helps us see what's happening.

For instance, at ( x = 3 ):

• The slope is ( 6 ).

This means the function is getting bigger really fast at that point!