Graphical+representation+of+derivative,+both+as+slope+and+as+functions

=Graphical Representation of Derivative=

Definition
Derivative - the limit of the ratio of the change in a function to the corresponding change in its independent variable as the latter change approaches zero (Merriam - Webster) You've just been sabotaged.-Clem

What does this mean?
Simply put, a derivative is the instantaneous rate of change of a function of f(x) with respect to x at some value x = c, or how much it changes in the vertical (y) direction for every step in the horizontal (x) direction at a certain value.

Numerically (as a function)
To find the derivative numerically, one must calculate with as much precision as possible the average rate of change of a function over as small an interval as possible.

While the rules of derivation won't all be covered, taking the derivative of a function basically means you will find a function that, for all values of x, represents the instantaneous rate of change of the main function.

Graphically (slope)
To find the derivative graphically, one must take a tangent line (at whatever point you are calculating the derivative) and find the slope of this tangent line.

This way of finding derivatives must be repeated for every value for which you are trying to find the instantaneous rate of change, as you don't actually get a function, but an individual straight line for every time the process is repeated.

Clarifications
f(x)=function f'(x)= derivative of function, AND slope of the function at any given point

Example 1:
f(x) = e^x f'(x) = e^x (in this case, the function and its derivative are the same, so they are graphically represented the same. http://www.intmath.com/Differentiation-transcendental/deriv-ex1.gif

For this graph, both the function AND its derivative function are represented by the blue curve. The tangent line at x = 2 is represented by the straight black line.

Example 2:
f(x)=x^2 f'(x)=2x (User created)

The "top", concave-up graph is the function. The "middle" graph, with a slope of 2, is the derivative function. The "bottom" graph, with a slope of 4, is the tangent line to the function at x = 2.

Notice that at x = 2, the "middle" graph has a value of f'(x) = 4.

Sources: http://en.wikipedia.org/wiki/Derivative http://mathworld.wolfram.com/Derivative.html http://www.merriam-webster.com/dictionary/derivative Calculus, Concepts and Applications, by Paul A. Foerster

Topic: Graphical Representation Group Members: Matt and Ben

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