Welcome back to Professor Baker's Math Class! In Chapter 3-2, we are moving beyond basic power rules to tackle functions that are multiplied together or divided by one another. If you've ever looked at a function like $y = xe^x$ or a rational function and wondered how to find the slope, this lesson is for you. We will introduce two powerhouse formulas: the Product Rule and the Quotient Rule.

1. The Product Rule

Intuition might tell us that the derivative of a product is simply the product of the derivatives, but as we showed in class with $f(x) = x^3$ and $g(x) = x^2$, that doesn't work! Instead, we use the Product Rule.

If $f$ and $g$ are both differentiable functions, then:

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

In simple terms: "Derivative of the first times the second, plus the first times the derivative of the second."

Example from Class Notes:
Let's differentiate $m(x) = xe^x$.

  • Let $f(x) = x \rightarrow f'(x) = 1$
  • Let $g(x) = e^x \rightarrow g'(x) = e^x$

Applying the rule:

$$ m'(x) = (1)(e^x) + (x)(e^x) = e^x(1+x) $$

2. The Quotient Rule

When functions are divided, we use the Quotient Rule. The order of operations is crucial here because of the subtraction in the numerator.

The formula is:

$$ \frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{g(x)f'(x) - f(x)g'(x)}{[g(x)]^2} $$

A popular mnemonic to remember this is: "Lo d-Hi minus Hi d-Lo, over Lo Lo" (where "Lo" is the denominator and "Hi" is the numerator).

Example:
Consider the function $y = \frac{e^x}{1+x^2}$. Using the rule, we keep the denominator squared at the bottom and expand the numerator:

$$ y' = \frac{(1+x^2)(e^x) - (e^x)(2x)}{(1+x^2)^2} $$ $$ y' = \frac{e^x(1 - 2x + x^2)}{(1+x^2)^2} = \frac{e^x(1-x)^2}{(1+x^2)^2} $$

3. Applications: Tangent Lines & Graphs

Remember, the primary purpose of finding a derivative is often to find the slope of the tangent line at a specific point. In the notes, we looked at the curve $y = \frac{1}{1+x^2}$, historically known as the Witch of Maria Agnesi.

To find the equation of the tangent line at $(-1, \frac{1}{2})$:

  1. Find the derivative $y'$ using the Quotient Rule.
  2. Plug in the x-value ($-1$) to get the numerical slope, $m$.
  3. Use Point-Slope Form: $y - y_1 = m(x - x_1)$.

We also explored how to use these rules graphically. Even without an algebraic equation, if you are given a graph of $F(x)$ and $G(x)$, you can find values like $P'(2)$ by reading the function heights and slopes directly from the grid.

Keep practicing these rules! They can be algebraically messy, but if you stay organized and write out your $f, f', g,$ and $g'$ terms separately before combining them, you'll do great.