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

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

We estimate derivatives when an exact formula is unavailable, too complex, or when we only have data points.

The limit definition provides the theoretical basis for estimating derivatives by using a small 'h' to approximate the limit.

Draw a tangent line at the point of interest and find its slope. This slope approximates the derivative at that point.

Symmetrical points provide a more balanced approximation of the instantaneous rate of change around the point of interest.

Units provide context and meaning to the numerical value of the derivative, indicating the rate of change of one quantity with respect to another.

Select points closest to the point where you're estimating the derivative, ideally symmetrically positioned around it.

The slope of the tangent line at a point on a curve is equal to the derivative of the function at that point.

Calculators and software can quickly compute numerical derivatives, providing accurate estimates, especially for complex functions.

A negative derivative indicates that the function is decreasing at that point.

The smaller the value of 'h' used in the estimation (approaching the limit), the more accurate the approximation of the derivative.

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.