Welcome back, students! In this session, we are advancing into two critical areas of calculus: Chapter 3.3, which covers the derivatives of trigonometric functions, and Chapter 3.4, which introduces the essential Chain Rule. These concepts are fundamental building blocks for everything that follows, so let's break down the key takeaways from our class notes.

3.3 Derivatives of Trigonometric Functions

We started by looking at the graphs of sine and cosine. By analyzing the slopes of the tangent lines, we visually proved a beautiful relationship between these functions. You must commit the following standard derivatives to memory, as they will appear frequently:

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

Pro Tip: Notice a pattern in the notes? Any trigonometric function starting with "co" (cosine, cotangent, cosecant) results in a negative derivative.

We also applied these concepts to physics. In Example 3 of the notes, we analyzed an object on a vertical spring with position $s = 4 \cos t$. By differentiating, we found the velocity equation:

$$v(t) = s'(t) = -4 \sin t$$

And by differentiating again, we found the acceleration:

$$a(t) = s''(t) = -4 \cos t$$

3.4 The Chain Rule

The Chain Rule is arguably the most powerful tool for differentiation. It allows us to differentiate composite functions—functions inside of other functions. If $y = f(g(x))$, the rule is:

$$ \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) $$

Think of this as an "Outside-Inside" rule: Differentiate the outer function (keeping the inside the same), then multiply by the derivative of the inside function.

Distinguishing Functions

In class, we looked at the difference between $\sin(x^2)$ and $\sin^2(x)$. It is crucial to identify which function is the "inner" function $u$:

  • For $y = \sin(x^2)$: The outer function is sine, and the inner is $x^2$.
    $y' = \cos(x^2) \cdot (2x) = 2x\cos(x^2)$
  • For $y = \sin^2(x) = (\sin x)^2$: The outer function is the power function ($u^2$), and the inner is sine.
    $y' = 2(\sin x) \cdot (\cos x) = 2\sin x \cos x$

Exponential Functions & General Bases

Finally, we extended our rules to exponential functions with bases other than $e$. As derived in the notes, for any base $b > 0$:

$$ \frac{d}{dx}(b^x) = (\ln b) \cdot b^x $$

This explains why the derivative of $e^x$ is just $e^x$ (since $\ln e = 1$), but the derivative of $2^x$ is $(\ln 2)2^x$.

Keep practicing these rules! The key to mastering the Chain Rule is recognizing the layers of the function. Good luck with the homework, and I'll see you in the next class!