Welcome back to Professor Baker's Math Class! In this lesson, we are bridging the gap between basic differentiation and more complex functional analysis. Chapter 3-3 focuses on two essential skills: memorizing the derivatives of trigonometric functions and mastering the Chain Rule.

1. The Cycle of Trigonometric Derivatives

One of the most fascinating patterns in Calculus is the relationship between sine and cosine. As we visualized in the class notes, the derivatives of sine and cosine follow a cycle of four. If you start with $\sin(x)$ and keep taking derivatives, the pattern repeats:

  • $\frac{d}{dx}(\sin x) = \cos x$
  • $\frac{d}{dx}(\cos x) = -\sin x$
  • $\frac{d}{dx}(-\sin x) = -\cos x$
  • $\frac{d}{dx}(-\cos x) = \sin x$ (and we are back to the start!)

This cycle is incredibly useful for finding higher order derivatives. For example, if we need to find the $33^{rd}$ derivative of $g(t) = 4\cos t$, we don't need to differentiate 33 times. We can look at the remainder when 33 is divided by 4 (which is 1). This tells us the answer corresponds to the first step in the cycle relative to cosine.

2. The Six Trigonometric Functions

Beyond sine and cosine, we derived the remaining functions using the Quotient Rule (specifically for $\tan(x) = \frac{\sin x}{\cos x}$). Here is the complete list you should commit to memory:

  • $\frac{d}{dx}(\sin x) = \cos x$
  • $\frac{d}{dx}(\cos x) = -\sin x$
  • $\frac{d}{dx}(\tan x) = \sec^2 x$
  • $\frac{d}{dx}(\cot x) = -\csc^2 x$
  • $\frac{d}{dx}(\sec x) = \sec x \tan x$
  • $\frac{d}{dx}(\csc x) = -\csc x \cot x$

3. The Chain Rule

The second half of this chapter introduces the Chain Rule, which allows us to find the derivative of composite functions. This is often described as the "Onion Method"—you must peel (differentiate) the outside layer first, leaving the inside alone, and then multiply by the derivative of the inside.

Formal Definition: If $y = f(g(x))$, then:

$$y' = f'(g(x)) \cdot g'(x)$$

Example from class:
Let's look at the function $f(x) = (9x^2 - 2x)^{30}$.
1. Outer Derivative: Bring the power down and subtract one: $30(9x^2 - 2x)^{29}$.
2. Inner Derivative: Find the derivative of the inside polynomial: $18x - 2$.
3. Combine: $f'(x) = 30(9x^2 - 2x)^{29} \cdot (18x - 2)$.

4. Putting It All Together

Real-world calculus problems rarely use just one rule. As seen in the final pages of our notes, you will encounter problems that combine the Product Rule and the Chain Rule. When tackling a function like $f(x) = 5x^3 \cdot (\frac{-3x^2+5}{3x+1})^{1/3}$, remember to take it step-by-step. Identify your $u$ and $v$ terms for the product rule first, and apply the chain rule only when differentiating the specific term that requires it.

Keep practicing these derivation patterns—they are the building blocks for everything coming next in Chapter 4!