Welcome to a pivotal week in our Calculus journey! In tonight's lesson, we are making a strategic adjustment to the schedule. We will be covering Section 3.1 and Section 3.2; however, I have decided to skip the exponential functions in Section 3.1 for now. We will revisit those concepts later when we discuss the Differentiation of Logarithms, allowing for a more cohesive understanding of those topics.

Section 3.1: The Definition of the Derivative

We begin by formalizing the concept of the derivative. While we will soon learn shortcuts, it is crucial to understand the "long way"—the limit definition. This definition connects the slope of a secant line to the slope of a tangent line.

The derivative of a function $f(x)$ is defined as:

$$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$

Key Concepts from Section 3.1:

  • Notation: You will see derivatives written in various ways, including $f'(x)$, $y'$, and Leibniz notation $\frac{dy}{dx}$.
  • Differentiability vs. Continuity: Remember, differentiability implies continuity, but continuity does not guarantee differentiability. A function is not differentiable at points where the graph has:
    • Corners or "cusps" (sharp turns)
    • Discontinuities (jumps or holes)
    • Vertical tangents

Section 3.2: Differentiation Rules

Once we understand the limit definition, we can unlock the "shortcuts." These rules allow us to find derivatives much faster and are the tools you will use most frequently moving forward.

The Power Rule:
For any real number $n$, if $f(x) = x^n$, then:

$$\frac{d}{dx}(x^n) = nx^{n-1}$$

Other Essential Rules:

  • Constant Multiple Rule: $\frac{d}{dx}[c \cdot f(x)] = c \cdot f'(x)$
  • Sum and Difference Rule: $\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)$
  • Derivatives of Sine and Cosine:
    • $\frac{d}{dx}(\sin x) = \cos x$
    • $\frac{d}{dx}(\cos x) = -\sin x$

Higher Order Derivatives

Finally, we discussed that since the derivative $f'(x)$ is itself a function, it can be differentiated again. This leads to the second derivative ($f''$), the third derivative ($f'''$), and so on. In physics, if $s(t)$ is position, $s'(t)$ is velocity, and $s''(t)$ is acceleration.

Be sure to review the attached notes and practice these rules until they become second nature. Happy calculating!