To understand derivatives using tangent lines, let’s break it down into simpler parts:

1. What is a Derivative?
   A derivative at a point shows us how steep the tangent line is to the function right at that point.

2. Tangent Line Equation:
   You can write the tangent line like this:
   $y = f(a) + f'(a)(x - a)$
   Here, $f'(a)$ is just the derivative when $x = a$.

3. Seeing it on a Graph:
   When we look at two points getting really close together, the line connecting them (called the secant line) becomes the tangent line. This helps us see the instant change in the function.

4. Why is this Important?
   Knowing about derivatives is crucial in calculus. It helps us understand how a function acts in a small area around a point.