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Summary

The slope m of a line given two points ( x 1 , y 1 ) and ( x 2 , y 2 ) is given by

m = y 2 − y 1 x 2 − x 1 .

Now we wish to study how to compute the slope of a general curve y = f ( x ) at a given point x = a . This is where the concept of secant lines comes in. A secant line is formed by connecting any two points ( a , f ( a ) ) and ( a + h , f ( a + h ) ) that lie on the curve y = f ( x ) , 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 h → 0 secant lines are created that contain the points ( a , f ( a ) ) and ( a + h , f ( a + h ) ) . These secant lines have slopes

m = f ( a + h ) − f ( a ) a + h − a = f ( a + h ) − f ( a ) h .

Thus as h → 0 the slopes of the secant lines are approaching the slope of the tangent line that contains the point x = a or ( a , f ( a ) ) . The slope of the tangent line at point x = a or ( a , f ( a ) ) is denoted as f ′ ( a ) and is calculated as follows:

f ′ ( a ) = lim h → 0 f ( a + h ) − f ( a ) h

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, f ′ ( a ) of a function f ( x ) 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 v ( t ) at time t is the derivative of the position s ( t ) at time t . The average velocity over a given interval [ a , b ] is given by

a v g v e l = s ( b ) − s ( a ) b − a

which can be interpreted as the slope of the secant line that contains the points ( a , s ( a ) and ( b , s ( b ) . Therefore, the instantaneous velocity at time t = a is given by

v ( a ) = s ′ ( a ) = lim h → 0 s ( a + h ) − s ( a ) h

We may estimate a derivative by using a table of values.

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