Welcome back to Professor Baker's Math Class! After spending time calculating derivatives using the long-form limit definition, we have finally arrived at the "shortcuts." In Chapter 3-1 and 3-2, we establish the fundamental rules of differentiation that will serve as the backbone for the rest of your calculus journey.
Below is a breakdown of the key concepts, formulas, and common pitfalls discussed in the lecture notes.
1. The Basic Building Blocks
The most commonly used rule you will encounter is the Power Rule. This allows us to differentiate polynomial terms instantly without setting up a limit.
The Power Rule: If $n$ is any real number, then:
$$\frac{d}{dx}(x^n) = nx^{n-1}$$Example from notes: If $f(x) = 2x^9$, then $f'(x) = 18x^8$.
Pro-Tip for Radicals and Fractions: Often, functions aren't written in a friendly $x^n$ format. You must rewrite them first using algebra rules:
- Radicals: Convert roots to fractional exponents (e.g., $\sqrt{x} = x^{1/2}$).
- Reciprocals: Move denominators to the numerator using negative exponents (e.g., $\frac{1}{x^5} = x^{-5}$).
2. Watch Out for Constants!
One of the trickiest parts of the notes involves identifying what is actually a constant. Remember, the derivative of a constant is always zero.
Consider the function $y = z^3 + e^3$.
- The derivative of $z^3$ is $3z^2$ (Power Rule).
- The derivative of $e^3$ is $0$. Why? Because $e^3 \approx 20.08$ is just a number, not a variable function!
3. The Natural Exponential Function
We also introduced the derivative of the natural exponential function, which is unique because it is its own derivative:
$$\frac{d}{dx}(e^x) = e^x$$4. The Product and Quotient Rules
When functions are multiplied or divided, we cannot simply take the derivative of each part separately. We must use specific rules.
The Product Rule
Used when differentiating two functions multiplied together, $f(x)g(x)$:
$$\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$$The Quotient Rule
Used when differentiating a fraction, $\frac{f(x)}{g(x)}$. A helpful mnemonic is "Lo d-Hi minus Hi d-Lo, over Lo Lo":
$$\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$$5. Applications: Tangent Lines and Horizontal Tangents
Differentiation isn't just about finding formulas; it's about analyzing slopes. We covered several application types:
- Tangent Line Equation: Find the derivative $f'(x)$, plug in the $x$-value to get the slope ($m$), and use point-slope form: $y - y_1 = m(x - x_1)$.
- Normal Line: The normal line is perpendicular to the tangent. Its slope is the negative reciprocal of the tangent slope ($m_{\perp} = -\frac{1}{m}$).
- Horizontal Tangents: To find where a curve flattens out, set the derivative equal to zero ($f'(x) = 0$) and solve for $x$.
6. Conceptual Understanding: Tables and Graphs
Finally, we practiced applying these rules without explicit equations. Whether you are given a table of values for $f(7)$ and $g'(7)$, or a piecewise linear graph where you calculate slope by looking at the rise-over-run, the rules remain the same.
Review the attached PDF for detailed handwritten solutions, including how to handle higher-order derivatives (like finding $g'''(t)$). Keep practicing these rules until they become second nature!