Lesson 1.4: Calculus Review - Derivatives (The 'Slope')

Welcome! In this lesson, we're going to learn about one of the two most important ideas in all of calculus: the Derivative.

This is more than just a math formula. It's a "superpower" that lets you measure instantaneous change. If you've ever looked at a car's speedometer and wondered, "How does it know my speed right now?", you were asking for a derivative.

The "Why": The Speedometer Problem

In finance, risk management is just another name for measuring "sensitivity to change." We must be able to answer "how-fast" questions:

How fast does my option price change when the stock price moves $1?
How fast does my portfolio lose value as time passes?
How fast does my bond price change when interest rates move 1%?

All of these are "speed" questions. The tool we use to answer them is the derivative.

The Core Paradox: How Do You Measure "Instantaneous"?

This is the central "wow" moment of all calculus. Our 7th-grade algebra formula for speed is:

Speed = Change in Distance / Change in Time

This formula requires a time interval (e.g., 1 hour). But a speedometer's "instantaneous speed" happens over a time interval of zero. This gives:

Speed = 0 / 0

...which is undefined! This is the paradox. How can we find the "speed" at a single, indivisible point in time?

The answer is the Derivative.

The Core Concept: Finding the Slope

Let's visualize this. In the "hard case" of a curved line like $f(x) = x^2$ , the slope is constantly changing. How can we find the exact slope at a single, instantaneous point (like $x=2$ )?

The "Calculus" Leap: The Limit (The "Cheat")

Here is the brilliant "cheat" that makes calculus work. We can't find the slope at one point, but we can find the slope of a line between two points.

Pick Two Points: We'll pick our "anchor point" $x$ and a second point that's just a tiny step $\\Delta x$ away, called $x + \\Delta x$ .
Find Their "Y" Values: The "y-value" at our anchor point is $f(x)$ , and at our new point is $f(x + \\Delta x)$ .
Calculate the "Fake" Slope: This gives us the slope of the secant line between the two points:

\text{Approximate Slope} = \frac{f(x + \Delta x) - f(x)}{\Delta x}

To make the approximation perfect, we make the "step" $\\Delta x$ infinitely small. We take the limit as $\\Delta x$ approaches 0. This "perfect" slope that it approaches is the Derivative.

Interactive Slope Explorer

Drag the slider to move the anchor point along the curve $f(x) = x^2$. Observe how the slope of the tangent line changes.

Anchor Point (x): 2.00

Slope at x = 2.00 is: 4.00

The "How-To": Common Rules

You don't need to re-derive these every time. Here are the "lookup table" rules we will use in this course.

Function $f(x)$	Derivative $f'(x)$	Example
$c$	$0$	The slope of a flat line is 0.
$x^n$	$n \cdot x^{n-1}$	$f(x)=x^3 \\implies f'(x)=3x^2$
$e^x$	$e^x$	The function's value is its slope!
$\\ln(x)$	$1/x$
$\\sin(x)$	$\\cos(x)$

Partial Derivatives (The Bridge to Lesson 1.7)

What if our function has two inputs, like our option price $V(t, S_t)$ ? This is the "Mountain Map" problem. A Partial Derivative is a derivative in one specific direction, while you pretend all other variables are just constant numbers. We use a "curly d" ( $\\partial$ ) to show it's a partial derivative.

For $f(x, y) = 3x^2 + 5y^4$ :

$\\frac{\\partial f}{\\partial x} = 6x$

$\\frac{\\partial f}{\\partial y} = 20y^3$

What's Next? (The 'Hook')

You've just learned what the "Greeks" are! In finance, this isn't just abstract math. It is the literal definition of risk:

Delta ( $\\Delta$ ) is just $\frac{\partial V}{\partial S}$ (the "slope" in the stock's direction).
Theta ( $\\Theta$ ) is just $\frac{\partial V}{\partial t}$ (the "slope" in the time direction).
Gamma ( $\\Gamma$ ) is the second derivative, $\frac{\partial^2 V}{\partial S^2}$ (the "curvature" or the "change in the slope").

This tool, the partial derivative, is the heart of risk management. Now that we can "slice" (Derivatives) and "sum" (Integrals), we are ready to build our most important tool: the Taylor Expansion (Lesson 1.6).

Lesson 0.3: The Normal Distribution N(μ, σ²)

Lesson 0.5: Calculus Review - Integrals (The "Sum")