Rules of Differentiation
Big Idea: Differentiation rules give us powerful shortcuts to find derivatives without using the limit definition every time. These rules let us break complex functions into simpler pieces.
📌 Recall: The Limit Definition
The derivative of $f(x)$ is defined as:
$$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$
While this definition is fundamental, applying it every time is tedious. The rules below are derived from this definition and make our work much faster.
Rule 0: Constant Rule & Power Rule (Foundation)
Constant Rule
If $f(x) = c$ where $c$ is a constant, then:
$$\frac{d}{dx}[c] = 0$$
Power Rule
If $f(x) = x^n$ where $n$ is any real number, then:
$$\frac{d}{dx}[x^n] = nx^{n-1}$$
Constant Multiple Rule
If $c$ is a constant and $f(x)$ is differentiable:
$$\frac{d}{dx}[c \cdot f(x)] = c \cdot f'(x)$$
You can "pull out" constants when differentiating.