AP Calculus BC / Unit 2: Differentiation: Definition and Fundamental Properties
2.2 Defining the Derivative of a Function and Using Derivative Notation
Notes
$$f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$$
- Alternate form at a point: $\displaystyle f'(a)=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}$.
- Notation: $f'(x)$, $y'$, $\displaystyle \frac{dy}{dx}$, $\displaystyle \frac{d}{dx}f(x)$.
Practice
$\displaystyle \lim_{h\to0}\frac{(3+h)^2-9}{h}$ equals:
It's $f'(3)$ for $f(x)=x^2$, which is $2(3)=6$.
$\displaystyle \lim_{x\to\pi}\frac{\sin x-\sin\pi}{x-\pi}$ equals:
It's the derivative of $\sin x$ at $\pi$: $\cos\pi=-1$.
Which notation does NOT mean the derivative of $y=f(x)$?
$\Delta y$ is a change in $y$, not a rate.
Ready for more? Take the Unit 2 test.