Derivatives & Tangents (College Board AP® Calculus BC): Revision Note

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 $f$ is defined by

  • $f^{\text{'}}\left(x\right)=\underset{h\to 0}{\lim }\frac{f\left(x+h\right)-f\left(x\right)}{h}$

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

  • Note that the derivative is also a function of x

• For example, $f\text{'}\left(2\right)$ represents the instantaneous rate of change of the function $f$ at the point $x=2$

• There are several ways to denote the derivative of $f\left(x\right)$

  • $f^{\text{'}}\left(x\right)$

  • $\frac{dy}{dx}$

  • $y^{\text{'}}$

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

  • They mean the same thing

How are derivatives and tangents related?

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

  • However, it may cut the curve somewhere else

• 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

  • For example, $f\text{'}\left(2\right)$ represents the slope of the tangent line to the graph $y=f\left(x\right)$ at the point $x=2$

When will tangent lines be horizontal or vertical?

• If the tangent line to the graph of a function $f$ is horizontal, then $f^{\text{'}}\left(x\right)=0$

  • Horizontal lines have a slope of zero

• If the tangent line to the graph of a function $f$ is vertical, then $f^{\text{'}}\left(x\right)$ is undefined

  • E.g. $f\left(x\right)=x^{\frac{1}{3}}$ with derivative $f^{\text{'}}\left(x\right)=\frac{1}{3x^{{\displaystyle \frac{2}{3}}}}$

    • $f^{\text{'}}\left(0\right)=\frac{1}{0}$, which is undefined, even though $f\left(x\right)$ is defined at $x=0$

    • Therefore the graph of $f$ has a vertical tangent at $x=0$

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

  • E.g. the derivative of $g\left(x\right)=\left|x\right|$ is undefined at $x=0$, because the left- and right-hand limits defining the derivative at that point are not equal

  • The graph of $g$ 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. $h\left(x\right)=\frac{1}{x}$ has a vertical asymptote at $x=0$

  • But the function is not defined when $x=0$, so it has no tangent at that point

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 $f$ at the point $\left(a,b\right)$ using a derivative:

  • Represent the equation of the tangent using the general form for the equation of a straight line with slope $m$ that goes through point $\left(x_1,y_1\right)$

    • $y-y_1=m\left(x-x_1\right)$

  • Substitute in $\left(a,b\right)$ as the point $\left(x_1,y_1\right)$

    • This is the point the tangent touches on the curve

  • Find the value of the derivative of $f\left(x\right)$ at the point $\left(a,b\right)$ if it is not given; this is $f^{\text{'}}\left(a\right)$

    • The value of the derivative of $f\left(x\right)$ at $\left(a,b\right)$ is equal to the slope of the tangent at $\left(a,b\right)$

  • Substitute in $f^{\text{'}}\left(a\right)$ as the value of $m$ in the equation of the tangent

  • The equation of the tangent will be of the form

    • $y-b=f^{\text{'}}\left(a\right)\left(x-a\right)$

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

    • For example, $y=b+f^{\text{'}}\left(a\right)\left(x-a\right)$