Derivatives
2.1 Definition of Derivative
Difference Quotient
The derivative at $x=a$ is defined by
$$ f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h} $$
Definition
If this limit exists, $f$ is differentiable at $a$.
Geometric Meaning
The derivative is the slope of the tangent line.
Example
For $f(x)=x^2$,
$$ f'(a)=\lim_{h\to 0}\frac{(a+h)^2-a^2}{h}=\lim_{h\to 0}(2a+h)=2a $$
Quick Notes
- Differentiable implies continuous.
- Continuous does not always imply differentiable.
Warning. The absolute value function is continuous at $0$ but not differentiable there.