Welcome back, class! In this session, we are bridging the gap between Algebra and Calculus by introducing one of the most powerful tools in mathematics: The Derivative. If you have ever wondered how to find the slope of a curved line or how to calculate the exact speed of a falling object at a specific split second, you are in the right place.

1. The Tangent Line Problem

In algebra, finding the slope of a straight line is easy ($m = \frac{rise}{run}$). But how do we find the slope of a curved function, like a parabola, at a single point? We use a Tangent Line.

As we discussed in the notes, we start with a secant line between two points and use a limit to bring those points infinitely close together. This gives us the formal definition of the slope of a curve at a point $a$:

$$m = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$$

Alternatively, using $h$ to represent the distance between points (where $h \to 0$), we get the definition used most frequently in Calculus I:

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

2. The Derivative Definition

This limit isn't just for geometry; it defines the Derivative of a function, denoted as $f'(x)$ (read as "f-prime of x"). We also saw other notations in class, such as $\frac{dy}{dx}$ or $\frac{df}{dx}$. Regardless of the notation, the process remains the same: we are calculating an instantaneous rate of change.

3. Algebraic Techniques for Finding Derivatives

Calculating these limits requires strong algebra skills. Based on the examples in our notes, here are the three main strategies you will need:

  • Polynomials (e.g., $y=x^2$): Expand the terms using FOIL or binomial expansion, subtract the original function, and cancel out the $h$ in the denominator.
  • Rational Functions (e.g., $y=3/x$): Find a common denominator to combine fractions in the numerator, then simplify complex fractions.
  • Radicals (e.g., $y=\sqrt{x}$): Multiply the numerator and denominator by the conjugate (e.g., $\sqrt{x+h} + \sqrt{x}$) to clear the square roots.

4. Physics Application: Instantaneous Velocity

One of the coolest applications of the derivative is in physics. If we have a position function $s(t)$, the derivative $s'(t)$ represents the instantaneous velocity, $v(t)$.

In our class example involving the CN Tower, we looked at a ball dropping with the position function $s = 4.9t^2$. By taking the derivative, we found the velocity function to be $v(t) = 9.8t$. This allows us to know exactly how fast the ball is traveling at any specific second, not just its average speed!

5. When Does a Derivative Not Exist?

Finally, remember that not all functions are differentiable everywhere. As shown in the graphs in our notes, a function fails to have a derivative at:

  • Corners or Cusps: Sharp turns (like $f(x) = |x|$ at $x=0$).
  • Discontinuities: Jumps or holes in the graph.
  • Vertical Tangents: Where the slope becomes undefined (vertical).

Keep practicing those limits, and don't let the algebra scare you! Understanding the concept of the "Difference Quotient" is key to mastering the rest of the semester.