Calc 1: Section 3-1 - Derivatives

Welcome to Section 3-1 of Calculus 1! This section introduces the fundamental concept of the derivative and explores several important rules that will help you calculate derivatives efficiently. We'll cover the power rule, the constant multiple rule, and the sum/difference rules. Let's dive in!

The Definition of the Derivative

The derivative of a function $f(x)$, denoted as $f'(x)$, represents the instantaneous rate of change of the function with respect to $x$. Formally, it's defined using a limit:

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

Let's consider the example from the notes. If $f(x) = x^3 - x$, find $f'(x)$:

$$f'(x) = \lim_{h \to 0} \frac{(x+h)^3 - (x+h) - (x^3 - x)}{h}$$ $$f'(x) = \lim_{h \to 0} \frac{x^3 + 3x^2h + 3xh^2 + h^3 - x - h - x^3 + x}{h}$$ $$f'(x) = \lim_{h \to 0} \frac{3x^2h + 3xh^2 + h^3 - h}{h}$$ $$f'(x) = \lim_{h \to 0} 3x^2 + 3xh + h^2 - 1 = 3x^2 - 1$$

Key Derivative Rules

Fortunately, we don't always have to use the limit definition to find derivatives. Several rules make this process much easier:

  • Derivative of a Constant Function: If $f(x) = c$, where $c$ is a constant, then $f'(x) = 0$. In other words: $$\frac{d}{dx}(c) = 0$$
  • The Power Rule: If $f(x) = x^n$, where $n$ is any real number, then $f'(x) = nx^{n-1}$. This is a cornerstone rule! $$\frac{d}{dx}(x^n) = nx^{n-1}$$ For example, if $f(x) = x^7$, then $f'(x) = 7x^6$. If $f(x)=\sqrt{x} = x^{\frac{1}{2}}$, then $f'(x) = \frac{1}{2}x^{-\frac{1}{2}} = \frac{1}{2\sqrt{x}}$.
  • The Constant Multiple Rule: If $c$ is a constant and $f$ is a differentiable function, then $$\frac{d}{dx}[cf(x)] = c \frac{d}{dx}f(x)$$
  • The Sum and Difference Rules: If $f$ and $g$ are both differentiable, then:
    • $$\frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}f(x) + \frac{d}{dx}g(x)$$
    • $$\frac{d}{dx}[f(x) - g(x)] = \frac{d}{dx}f(x) - \frac{d}{dx}g(x)$$

Examples

Let's use these rules to find some derivatives:

  1. If $f(x) = 5x^6$, then $f'(x) = 5(6x^5) = 30x^5$.
  2. If $g(x) = \frac{3}{4}x^{\frac{5}{4}}$, then $g'(x) = \frac{3}{4}(\frac{5}{4}x^{\frac{1}{4}}) = \frac{15}{16}x^{\frac{1}{4}}$.
  3. If $f(x) = 5x^2 + x - 3$, then $f'(x) = 5(2x) + 1 - 0 = 10x + 1$. Then the second derivative is $f''(x) = 10$ and the third derivative is $f'''(x) = 0$.
  4. If $f(x) = x^5 - 7x^3 + x^2 - 5$, then $f'(x) = 5x^4 - 21x^2 + 2x$.
  5. If $g(m) = \frac{1}{m} + \frac{1}{m^2} = m^{-1} + m^{-2}$, then $g'(m) = -1m^{-2} + (-2)m^{-3} = -\frac{1}{m^2} - \frac{2}{m^3}$.
  6. If $h(w) = \sqrt{2w} - \sqrt{2}$, then $h'(w) = \sqrt{2} \cdot 1 - 0 = \sqrt{2}$.
  7. If $h(x) = \pi x^3 + 5x^2$, then $h'(x) = 3\pi x^2 + 10x$.
  8. If $m(x) = (3x^2 - 2)(5x + 1) = 15x^3 + 3x^2 - 10x - 2$, then $m'(x) = 45x^2 + 6x - 10$.

Keep practicing, and you'll master these rules in no time. Good luck!