Welcome back to class! In Section 3-1, we are making a massive leap forward in our Calculus journey. Up until now, we have been calculating derivatives using the long, rigorous limit definition. While that is essential for understanding the why, today we learn the how—the shortcut rules that make differentiation much faster and more efficient.

Here is a breakdown of the key rules and concepts covered in the lecture notes:

1. The Derivative of a Constant Function

Let's start with the simplest rule. If you have a constant function (a horizontal line), its slope never changes—it is always zero. Therefore:

$$\frac{d}{dx}(c) = 0$$

2. The Power Rule

This is likely the rule you will use most often. If you have a variable raised to a power $n$, you simply bring the power down to the front and subtract one from the exponent:

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

Examples from class:

  • If $f(x) = x^6$, then $f'(x) = 6x^5$.
  • If $y = t^4$, then $y' = 4t^3$.

Important Algebraic Note: The Power Rule works for negative and fractional exponents too! You must be comfortable converting roots to fractional powers and fractions to negative exponents before differentiating.

  • $\frac{1}{x^2} = x^{-2} \rightarrow$ Derivative is $-2x^{-3}$ or $\frac{-2}{x^3}$
  • $\sqrt{x} = x^{1/2} \rightarrow$ Derivative is $\frac{1}{2}x^{-1/2}$ or $\frac{1}{2\sqrt{x}}$

3. Linearity Rules (Sums and Constant Multiples)

Calculus is "linear," which means we can differentiate term by term. If you have a polynomial, you just take the derivative of each piece separately. You can also pull constant multipliers out front.

$$\frac{d}{dx}[cf(x)] = c \frac{d}{dx}f(x)$$

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

4. The Natural Exponential Function

We introduced a very special function, $e^x$. It is unique in mathematics because the slope of the curve at any point is equal to the value of the function itself. Its derivative is itself!

$$\frac{d}{dx}(e^x) = e^x$$

5. Applications: Physics and Tangent Lines

We applied these new rules to two major concepts:

  • Tangent and Normal Lines: We use the derivative to find the slope ($m$) of the tangent line at a specific point, then use the point-slope form $(y - y_1) = m(x - x_1)$ to write the equation. Remember, the normal line is perpendicular, so its slope is the negative reciprocal ($-1/m$).
  • Physics of Motion: If you have a position function $s(t)$, the derivative is velocity $v(t)$, and the derivative of velocity is acceleration $a(t)$.
    $$s(t) \xrightarrow{\text{derive}} v(t) \xrightarrow{\text{derive}} a(t)$$

Remember to review the attached notes for detailed examples, especially on how to simplify complex algebraic expressions before taking the derivative. Keep practicing those exponent rules!