Secant: intersects curve at two points | Tangent: touches curve at one point, represents derivative.

Average: slope of secant line | Instantaneous: slope of tangent line, the derivative.

By hand: uses limit definition or nearby points, approximate | Technology: uses algorithms, more accurate.

Symmetric: more accurate approximation | Non-symmetric: less accurate, but still useful.

Table: Uses discrete data points to calculate slope | Graph: Requires drawing a tangent line and estimating its slope visually.

Larger 'h': Less accurate estimation | Smaller 'h': More accurate estimation, approaching the true derivative.

Linear: Derivative is constant, easy to estimate | Non-linear: Derivative varies, requires more careful estimation.

Limit definition: fundamental, works for all functions | Power rule: shortcut, only applies to power functions.

Calculators: Portable, good for quick calculations | Desmos: Visual, good for graphing and exploring functions.

Secant lines: approximate average rate of change | Tangent lines: approximate instantaneous rate of change.

$f'(a) = \lim_{{h \to 0}} \frac{{f(a + h) - f(a)}}{h}$

$f'(a) \approx \frac{f(a + h) - f(a)}{h}$ for a small h

$f'(x) \approx \frac{f(x_2) - f(x_1)}{x_2 - x_1}$

$f'(a) \approx \frac{f(a + h) - f(a - h)}{2h}$

$f'(2.25) \approx \frac{f(2.5) - f(2)}{2.5 - 2}$

$\frac{d}{dx}(f(x))|_{x=a}$

Choose two points close to the desired point. Use the slope formula: $\frac{\Delta y}{\Delta x}$

Draw a tangent line at the point of interest and calculate its slope: $\frac{rise}{run}$

$\frac{f(x_2) - f(x_1)}{x_2 - x_1}$

$\Delta x [f(x_1) + f(x_2) + ... + f(x_n)]$, where $x_i$ are the midpoints of the subintervals.

A large magnitude derivative, indicating a rapid rate of change.

A derivative of zero, indicating a stationary point (local max/min).

Positive: increasing, Negative: decreasing, Zero: horizontal tangent.

Look for sharp corners, cusps, or vertical tangent lines.

It represents the process of finding the instantaneous rate of change as the interval shrinks to zero.

Draw a tangent line at the point and determine its slope.

The function is increasing at that point.

The function is decreasing at that point.

A point where the original function has a horizontal tangent (critical point).

Look for the steepest part of the graph where the tangent line would have the greatest slope magnitude.