Welcome to Unit 2: The Heart of Calculus!
In Unit 1, we talked about limits. Now, we are going to use those limits to do something amazing: find the exact steepness of a curve at a single point. This is called differentiation. Differentiation is simply a fancy word for finding the rate of change. Whether you are tracking the speed of a car, the growth of a population, or the changing price of a stock, you are using the concepts of Unit 2. Don't worry if this seems tricky at first—calculus is like a new language, and we are going to learn it one word at a time!
2.1 & 2.2: Defining the Derivative
In algebra, you learned how to find the slope of a straight line using two points: \( m = \frac{y_2 - y_1}{x_2 - x_1} \). But what if you have a curve and you only have one point? This is where the derivative comes in.
The Core Concept: The derivative of a function at a point is the instantaneous rate of change. It is the slope of the tangent line (a line that just grazes the curve at that specific point).
The Limit Definition of a Derivative
To find that slope at a single point, we imagine picking a second point very, very close to the first one and then moving it until the distance between them is basically zero. We use this formula:
\( f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \)
Think of it this way: The \( h \) represents a tiny "nudge" away from your point. As that nudge gets smaller and smaller (\( h \to 0 \)), your average slope becomes the instantaneous slope.
Derivative Notation
You will see the derivative written in several ways. They all mean the same thing:
1. \( f'(x) \) (Read as "f prime of x")
2. \( \frac{dy}{dx} \) (Read as "the derivative of y with respect to x")
3. \( \frac{d}{dx} [f(x)] \) (An instruction to "take the derivative" of what follows)
Key Takeaway: The derivative is just a formula for the slope of a function at any given point.
2.3: Estimating Derivatives
Sometimes you don't have an equation; you only have a table of values. To estimate the derivative at a point, just find the average slope between the two closest points available.
Example: If you want to estimate the slope at \( x = 3 \), and you have data for \( x = 2 \) and \( x = 4 \), use the old-school slope formula: \( \frac{f(4) - f(2)}{4 - 2} \).
2.4: Differentiability and Continuity
For a derivative to exist at a point, the function must be differentiable there. There is a very important rule to remember:
If a function is differentiable, it MUST be continuous.
However, just because a function is continuous (no holes or gaps) doesn't mean it is differentiable. A function fails to have a derivative at a point if:
1. It has a sharp turn or "corner" (like the bottom of a V in an absolute value graph).
2. It is not continuous (there is a hole, jump, or asymptote).
3. It has a vertical tangent (the slope becomes infinitely steep).
Analogy: Think of a roller coaster. For it to be a "smooth" ride (differentiable), the track can't have any breaks (continuity), but it also can't have any sharp jagged edges that would snap the car off the tracks.
Quick Review: Differentiability implies Continuity. (D \(\to\) C). But Continuity does not always imply Differentiability.
2.5, 2.6 & 2.7: The Basic Rules of Differentiation
Good news! You don't have to use the long limit definition every time. We have shortcuts!
The Power Rule
This is the most used rule in calculus. If you have \( f(x) = x^n \), then:
\( f'(x) = nx^{n-1} \)
The Trick: "Bring the power down to the front, then subtract one from the exponent."
Example: The derivative of \( x^3 \) is \( 3x^2 \).
The Constant Multiple and Sum/Difference Rules
1. Constant Rule: The derivative of a plain number (like 5) is always 0. (Because a flat horizontal line has no slope!)
2. Constant Multiple: The derivative of \( 5x^3 \) is \( 5 \cdot (3x^2) = 15x^2 \). The constant just "hangs out."
3. Sum/Difference: You can take the derivative of each part of an addition or subtraction problem separately.
Common Derivatives to Memorize
You need to know these by heart:
\( \frac{d}{dx} (\sin x) = \cos x \)
\( \frac{d}{dx} (\cos x) = -\sin x \) (Notice the negative sign!)
\( \frac{d}{dx} (e^x) = e^x \) (The easiest one—it never changes!)
\( \frac{d}{dx} (\ln x) = \frac{1}{x} \)
Did you know? The function \( e^x \) is the only non-zero function that is its own derivative. It is the "cool kid" of calculus because it stays exactly the same no matter how many times you differentiate it!
2.8 & 2.9: The Product and Quotient Rules
When functions are multiplied or divided, you cannot just take the derivative of each part. You must use these specific "recipes."
The Product Rule (For \( f \cdot g \))
\( \frac{d}{dx} [f(x)g(x)] = f'(x)g(x) + f(x)g'(x) \)
Mnemonic: "Left d-Right plus Right d-Left."
(The first function times the derivative of the second, plus the second function times the derivative of the first.)
The Quotient Rule (For \( \frac{High}{Low} \))
\( \frac{d}{dx} [\frac{f(x)}{g(x)}] = \frac{g(x)f'(x) - f(x)g'(x)}{[g(x)]^2} \)
Mnemonic: "Low d-High minus High d-Low, over Low-Low."
1. Low: The bottom function.
2. d-High: Derivative of the top.
3. High: The top function.
4. d-Low: Derivative of the bottom.
5. Low-Low: The bottom function squared.
Common Mistake: Forgetting the minus sign in the Quotient Rule or mixing up the order. Always start with the bottom function (Low)!
2.10: Derivatives of Other Trig Functions
Once you know the Quotient Rule, you can find the derivatives of the other four trig functions. You should memorize these to save time:
\( \frac{d}{dx} (\tan x) = \sec^2 x \)
\( \frac{d}{dx} (\cot x) = -\csc^2 x \)
\( \frac{d}{dx} (\sec x) = \sec x \tan x \)
\( \frac{d}{dx} (\csc x) = -\csc x \cot x \)
Memory Aid: Notice that the derivative of every "co-" function (\( \cos, \cot, \csc \)) starts with a negative sign. If it starts with "co," it’s negative!
Key Takeaway for Unit 2: Calculus is about patterns. Once you learn the Power Rule, Product Rule, and Quotient Rule, you can find the slope of almost any function you'll ever encounter. Keep practicing these rules until they feel like second nature!