APCALCDerivatives

Derivatives  The derivative of a function is a basic part of Calculus (particularly [|differential calculus]). But what is it? Where did it come from? And how can we find it? a  What is a derivative? A derivative in calculus is the slope of the tangent line to any point on a graph, or the instantaneous rate of change of a function. When finding the equation of tangent lines, the slope of the curve f(//x//) at (//a//, f(//a//)) is equal to the derivative of f at //a//. When finding rates of change, the instantaneous rate of change of a function f(//x//) when //x//=//a// is equal to the derivative of f at //a//. First, we need to know what a tangent is. A tangent line is one that touches an area of the graph at only one point, and does not intersect it. An example of a tangent line is shown in the graph below. On the graph of a [|non-linear function], like the one shown to the left, the slope varies. The slope of the tangent line tells us the slope of the function that is graphed at a particular point, and it also tells us what the rate of change is between two points. By finding the derivative of a function, we are essentially finding the slope of the function at any given point, aka the slope of the tangent line. a  a   a   In short, the derivative tell us:
 * the slope of a tangent to a curve or function at any given point
 * the [|rate of change] of one quantity compared to another

The function of a derivative is given as

or  ﻿The derivative can be written as    Where did it come from? The discovery of calculus itself is attributed to two mathematicians: [|Isaac Newton] and [|Gottfried Wilhelm von Leibniz]. Newton was the man who thought of x and y values varying over time, which lead to the idea of finding derivatives. However, Leibniz found that by taking the derivative of a function, one could find the slope of it's tangent lines as well as find it's rate of change. a  a   a   How can we find it? Given a graph below, we wish to find the derivative at point P, or the slope of the tangent line at P.

We can approximate the derivative by taking a point near P(x, f(x)), like Q(x+h, F(x+h))



The value of, from the graph above, is an approximation to the slope of the tangent. The slope is the change of y over the change of x, or.

If we move point Q closer to point P, the line PQ will get closer to the tangent line at point P. Therefore, the slope of PQ gets closer to the slope of the tangent line at point P.

If we let Q equal P, and therefore the distance between the two points (h) would be zero, we could find the exact slope of the tangent. Now, can be written as. The slope of PQ is given as. However, we want the slope at P, so we let h --> 0 (h approach 0). By that account, Q will approach P and will approach the slope that we are trying to obtain. Therefore, the slope of the tangent line, or the derivative, at point P can be given as:. We now have the [|instantaneous rate of change] of y with respect to x, and the function for the derivative.

Is it always possible to take the derivative? There are some times, however, where taking the derivative is not possible. This is when the function is not [|differentiable].

A function is differentiable at //a// if f'(//a//) exists, when a is a point on the x-axis. It is differentiable on the open [|interval] (//a//, //b//) if it is differentiable at every number in the interval. If a function is differentiable, it is also [|continuous]. The [|contrapositive] of this theorem states that if a function is [|discontinuous] at //a// then it is not differentiable at //a//.

The following graphs show examples of functions that are not differentiable.

This function is not differentiable because there is [|discontinuity]. This function is not differentiable because it has a [|vertical tangent]. This function is not differentiable because it has a corner/cusp.

Are there special rules for derivatives? Rules exist in taking the derivative of an algebraic function, rather than a graphic function. They make taking the derivative much easier.

When taking the derivative of a [|constant], the result is always zero. When taking the derivative of a function with addition, we can use the [|sum rule]. When taking the derivative of a function with subtraction, we can use the [|difference rule]. When a function is raised to a power, we can use the [|power rule]. When there is multiplication in a function, we can use the [|product rule]. When there is division, we can use the [|quotient rule]. When we are given a function inside of another function, we can use the [|chain rule].

media type="custom" key="9395602" Derivatives media type="custom" key="9395604"Chain Rule

Below are the derivatives of trigonometric functions. More﻿ Information: By taking the derivative of a function, we learn more than just the slope of the tangent line or the rate of change. When we take the [|first derivative], we can tell whether the original function is increasing or decreasing on certain intervals. We do this by setting the derivative equal to zero and solving for x to find the critical points. The critical points tell us that there is a local maximum or minumum at those points. We then use those points to make specific intervals. If the derivative is positive in an interval, the direction of the original function is positive in that interval. If the derivative is negative in an interval, the direction of the original function is negative in the interval.

You can also take the derivative more than once. By taking the [|second derivative], we can find inflection points from it's zeroes and, by using the intervals, find where the orignial function in concave up or concave down. Where the second derivative is positive, the original function is concave up.

When both the first derivative and second derivative are positive on an interval, the original function is increasing. When both are negative on an interval, the original function is decreasing.

In Bri ﻿ ef: provided this limit exists. If this limit exists for each **x** in an open interval **I**, then we say that **f is differentiable on I**. ||  ||
 * || **Definition.** Let **y = f(x)** be a function. The **derivative of f** is the function whose value at x is the limit

and the **left-hand derivative of f at x = a** is the limit The function **f is differentiable on the interval I** if
 * || **Definition.** Let **y = f(x)** be a function and let **a** be in the domain of **f**. The **right-hand derivative of f at x = a** is the limit
 * when **I** has a right-hand endpoint **a**, then the left-hand derivative of **f** exists at **x = a**,
 * when **I** has a left-hand endpoint **b**, then the right-hand derivative of **f** exists at **x = b**, and
 * **f** is differentiable at all other points of **I**. ||  ||

Works Cited [] [] [] [] [] [] [] []