Welcome to Chapter 3! If you have been struggling with the long limit definition of the derivative ($ f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h} $), I have great news for you. We have finally arrived at the point in Calculus where we learn the Differentiation Rules. These are the shortcuts that allow us to find derivatives in seconds rather than minutes.

1. The Constant Rule

Let’s start with the simplest function: a horizontal line. If $ f(x) = c $ (where $c$ is a constant number), the graph is a horizontal line. What is the slope of a horizontal line? It's zero. Therefore:

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

Whether $ y=3 $, $ y=5 $, or $ y = \pi $, the derivative is always 0.

2. The Power Rule

This is the tool you will use most often in this course. Instead of expanding binomials like $(x+h)^4$, we can simply follow this pattern:

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

How it works: Bring the exponent down to the front as a multiplier, and then subtract 1 from the exponent.

  • If $ f(x) = x^4 $, then $ f'(x) = 4x^3 $.
  • If $ f(x) = x^7 $, then $ f'(x) = 7x^6 $.

3. Algebra First, Calculus Second

A major theme in our class notes is that the Power Rule applies to any real number exponent, but the function doesn't always look like $x^n$ right away. You must rewrite the function first.

Negative Exponents (Fractions):
If $ f(x) = \frac{1}{x^2} $, rewrite it as $ x^{-2} $.
Then differentiate: $ f'(x) = -2x^{-3} = \frac{-2}{x^3} $.

Rational Exponents (Roots):
If $ f(x) = \sqrt{x} $, rewrite it as $ x^{1/2} $.
Then differentiate: $ f'(x) = \frac{1}{2}x^{-1/2} = \frac{1}{2\sqrt{x}} $.

4. Sums, Differences, and Polynomials

Because the derivative is a linear operator, we can take the derivative of each term in a polynomial separately. We also use the Constant Multiple Rule, which says constants attached to variables "come along for the ride."

Example from class:

$$ \frac{d}{dx}(x^8 + 12x^5 - 4x^4 + 10x^3 - 6x + 5) $$

Apply the power rule to each piece:

$$ 8x^7 + 60x^4 - 16x^3 + 30x^2 - 6 $$

5. The Exponential Function

We also introduced the number $e$. The function $ f(x) = e^x $ is unique in mathematics because it is its own derivative. It models growth where the rate of change is proportional to the current amount.

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

6. Physics Application: Position, Velocity, Acceleration

Derivatives describe rates of change. In physics:

  • Position ($s$): Where the object is.
  • Velocity ($v$): The derivative of position ($s'$).
  • Acceleration ($a$): The derivative of velocity ($v'$), or the second derivative of position ($s''$).

Example: Given $ s = 2t^3 - 5t^2 + 3t + 4 $, find the acceleration at $t=2$.
1. Find velocity: $ v = s' = 6t^2 - 10t + 3 $
2. Find acceleration: $ a = v' = 12t - 10 $
3. Evaluate at $t=2$: $ a(2) = 12(2) - 10 = 14 \text{ cm/s}^2 $.

Make sure to review the examples in the notes where we simplify complex fractions before deriving. Practice these rules until they become second nature!