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Additional Instructional Resources
Defining the Derivative- DM1
Summary
The slope of a line given two points and is given by
Now we wish to study how to compute the slope of a general curve at a given point . This is where the concept of secant lines comes in. A secant line is formed by connecting any two points and that lie on the curve , and the slope of the secant line is calculated as above.
Since we know how to compute slopes of lines we use this information to compute the slope of a general curve. Thus as secant lines are created that contain the points and . These secant lines have slopes
Thus as the slopes of the secant lines are approaching the slope of the tangent line that contains the point or . The slope of the tangent line at point or is denoted as and is calculated as follows:
See the Desmos demonstration on Secant and Tangent Lines.
The slope of the tangent line to a curve measures the instantaneous rate of change of a curve. The derivative, of a function at a value a is found using the definition for the slope of the tangent line. Velocity is the rate of change of position. As such, the velocity at time is the derivative of the position at time . The average velocity over a given interval is given by
which can be interpreted as the slope of the secant line that contains the points and . Therefore, the instantaneous velocity at time is given by
We may estimate a derivative by using a table of values.