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A derivative can be used to describe the rate of change at a single point, or the instantaneous velocity.
The formula used to calculate the average rate of change looks like this:
\dfrac{f(b) - f(a)}{b - a}
Interpreting it, this can be described as the change in y over the change in x.
- Speed is always positive
- Velocity is directional
As the distance between the two points a and b grow smaller, we get closer and closer to the instantaneous velocity of a point. Limits are suited to describing the behavior of a function as it approaches a point.
If we have the coordinate pair (a, f(a)), and the value h is the distance between a and another x value, the coordinates of that point can be described as ((a + h, f(a + h)). With this info:
- The slope of the secant line can be described as
\dfrac{f(a + h) - f(a)}{a + h - a}, which simplifies to\dfrac{f(a + h) - f(a)}{h}. - The slope of the tangent line or the instantaneous velocity can be found by taking the limit of the above function as the distance (
h) approaches zero:
\lim_{h \to 0}\dfrac{f(a + h) - f(a)}{h}
The above formula can be used to find the derivative. This may also be referred to as the instantaneous velocity, or the instantaneous rate of change.
Line Types
Secant Line
A Secant Line connects two points on a graph.
A Tangent Line represents the rate of change or slope at a single point on the graph.
Notation
Given the equation y = f(x), the following are all notations used to represent the derivative of f at x:
f'(x)\dfrac{d}{dx}f(x)y'\dfrac{dy}{dx}\dfrac{df}{dx}- "Derivative of
fwith respect to $x$"
Functions that are not differentiable at a given point
- Where a function is not defined
- Where a sharp turn takes place
- If the slope of the tangent line is vertical
Higher Order Differentials
- Take the differential of a differential
Using the definition of a derivative to determine the derivative of f(x) = x^n, where n is any natural number.
f'(x) = \lim_{h \to 0} \dfrac{(x + h)^n - x^n}{h}
- Using pascal's triangle, we can approximate
(x + h)^n