Recall that the derivative of a function f at a point x = a is defined to be[br][math]f'\left(a\right)=lim_{h\longrightarrow0}\frac{f\left(a+h\right)-f\left(a\right)}{h}[/math].[br]When this limit exists it produces a single number for the choice of f and a that represents the slope of the tangent line to the curve at (a, f(a)). It is also the instantaneous rate of change in the function at that point.[br]
What happens if we choose a different value for a? Does the derivative change? Can you ever get two different values from the derivative?
Different choices for the value of a give different values for f '(a). However, for any specific choice of f and a there is only one single value for f '(a). Recall that this is exactly what we mean by a function.
If we think of a as a specific number, then f '(a) is also a specific number. However, if we replace a by x, and we interpret x as an arbitrary input then f '(x) is the corresponding variable output. Therefore, f '(x) is not a number, but rather a formula for a function.[br][br][math]f'\left(x\right)=lim_{h\longrightarrow0}\frac{f\left(x+h\right)-f\left(x\right)}{h}[/math]
Enter the formula for the original function that you want to investigate in the input box for f(x). Its graph will appear in blue.[br][br]Pick a value for a by moving the point on the x-axis, moving the slider, or typing in a value in the input box for a. You will see (a, f(a)) plotted on the graph.[br][br]Checking the Tangent Line checkbox will show the graph of the tangent line at (a, f(a)). Remember that the slope of this tangent line is the value of the derivative at that point.[br][br]Checking the Tangent Line Segment checkbox will show a little piece of the tangent line. It extends horizontally 0.5 units to the left and right of (a, f(a)). A horizontal line segment of length 1 is also displayed. [br]The vertical change of this line segment is the value of the derivative at x = a. This is illustrated by the vertical vector.[br][br]Checking the f'(a) box shows the value of f '(a) illustrated by a vertical vector starting on the x-axis at (a, 0). The end of the vector is the point (a, f '(a)), which is a point on the derivative function.[br][br]Experiment with different values of a.[br][br]Checking the Trace of (a, f '(a)) will trace this point as a is changed. Uncheck and move the graph a bit to clear the trace.[br][br]Checking Show Graph of f '(x) will show the graph of the derivative function.[br][br]Experiment by checking and unchecking boxes as desired, moving a, and changing the formula for the function.[br][br]
When the value of the derivative is positive, the graph of the derivative is above the x-axis. When the derivative is positive what can we say about the graph of the original function?[br][br]When the value of the derivative is negative, the graph of the derivative is below the x-axis. When the derivative is negative what can we say about the graph of the original function?[br][br]When the derivative is zero, the graph of the derivative is on the x-axis. When the derivative is zero what can we say about the graph of the original function?
When the derivative is positive, the slope of the original function is positive and the graph of the original function is increasing at that point.[br][br]When the derivative is negative, the slope of the original function is negative and the graph of the original function is decreasing at that point.[br][br]When the derivative is zero, the original function has a horizontal tangent line. [br]If a function has a local maximum or minimum then at that point either the derivative is zero, the derivative is undefined, or the function is at an endpoint.