Derivatives & Tangents (College Board AP® Calculus AB)

Derivatives and tangents

What is the derivative of a function?

• The derivative of a function describes the instantaneous rate of change of a function at any given point

  • It is equal to the slope of the curve at that point

• The derivative of the function is defined by

  • 

  • This is only valid for values of x where this limit exists

  • Note that the derivative is also a function of x

• There are several ways to denote the derivative of

  • 

  • 

  • 

• You may also see "the derivative of ..." written as " ... differentiated"

  • They mean the same thing

How are derivatives and tangents related?

• The value of the derivative of a function at a point is equal to the slope of the tangent to the graph at that point

  • A tangent to a curve is a line that just touches the curve at one point but doesn't cut it at or near that point

    • However it may cut the curve somewhere else

How do I find the equation of a tangent to a curve using a derivative?

• To find the equation of a tangent line to the graph of a function at the point using a derivative:

  • Represent the equation of the tangent using the general form for the equation of a straight line with slope that goes through point

    • 

  • Substitute in as the point

    • This is the point the tangent touches on the curve

  • Find the value of the derivative of at the point if it is not given; this is

    • The value of the derivative of at is equal to the slope of the tangent at

  • Substitute in as the value of in the equation of the tangent

  • The equation of the tangent will be of the form

    • 

  • This equation can then be rearranged to another desired form if needed

    • For example,

When will tangent lines be horizontal or vertical?

• The tangent line to the graph of a function will be horizontal when

  • Horizontal lines have a slope of zero

• The tangent line to the graph of a function will be vertical when is undefined because of dividing a constant by zero

  • E.g. with derivative

    • , which is undefined, even though is defined at

    • Therefore the graph of has a vertical tangent at

  • But be careful, as there are other reasons a derivative might not be defined at a point

    • E.g. the derivative of is undefined at , because the left- and right-hand limits defining the derivative at that point are not equal

    • The graph of does not have a vertical tangent (or any tangent) at that point

  • Also don't confuse vertical tangents with vertical asymptotes

    • Tangents and curves intersect, but curves only approach asymptotes without ever intersecting with them

    • E.g. has a vertical asymptote at

    • But the function is not defined when , so it has no tangent at that point