In this lesson, we will explore some interesting uses of derivatives. We will focus on tangent lines, how things change instantly, and solving optimization problems.

Tangent Lines
Tangent lines are important in calculus. The slope of a tangent line to a curve at a certain point is found using the derivative of the function at that point.

If we have a function called ( f(x) ), the slope of the tangent line at ( x = a ) can be figured out with this formula:

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

This idea helps us see how functions act nearby and is useful in areas like physics and engineering.

Instantaneous Rate of Change
The instantaneous rate of change shows us how a function changes at a specific point. This rate is represented by the derivative.

For example, if ( s(t) ) tells us where an object is at time ( t ), then the instantaneous rate of change, which is the object's speed, can be shown by:

$v(t) = s'(t)$

Understanding this concept is important in many fields, like business and biology, where change happens all the time.

Optimization Problems
Derivatives can also help us solve optimization problems. When we find critical points—where ( f'(x) = 0 )—we can identify the highest and lowest values of a function. This is key for real-world issues like reducing costs or increasing profits.

During our class discussion, we’ll look at how these derivative uses appear in different fields. This will help you understand better and learn more deeply.