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Section1.8Differentiability

Motivating Questions
  • What does it mean graphically to say that a function \(f\) is differentiable at \(x = a\text{?}\) How is this connected to the function being locally linear?

  • How are the characteristics of a function having a limit, being continuous, and being differentiable at a given point related to one another?

In this section we aim to determine whether or not the function has a derivative \(f'(a)\) at \(x = a\text{.}\)

SubsectionBeing Differentiable at a Point

We recall that a function \(f\) is said to be differentiable at \(x = a\) if \(f'(a)\) exists. Moreover, for \(f'(a)\) to exist, we know that the graph of \(y = f(x)\) must have a tangent line at the point \((a,f(a))\text{,}\) since the value of \(f'(a)\) is precisely the slope of this line. Observe that in order to ask if \(f\) has a tangent line at \((a,f(a))\text{,}\) it is necessary for \(f\) to be continuous at \(x = a\text{:}\) if \(f\) fails to have a limit at \(x = a\text{,}\) if \(f(a)\) is not defined, or if \(f(a)\) does not equal the value of \(\lim_{x \to a} f(x)\text{,}\) then it doesn't make sense to talk about a tangent line to the curve at this point.

Indeed, it can be proved formally that if a function \(f\) is differentiable at \(x = a\text{,}\) then it must be continuous at \(x = a\text{.}\) Stated differently, if \(f\) is not continuous at \(x = a\text{,}\) then it is automatically the case that \(f\) is not differentiable there. For example, in Figure1.97 below, both \(f\) and \(g\) fail to be differentiable at \(x = 1\) because neither function is continuous at \(x = 1\text{.}\)

Figure1.97Functions \(f\) and \(g\) that are not continuous (and hence not differentiable) at \(x=1\text{,}\) and a function \(h\) that is both continuous and differentiable at \(x=1\text{.}\)

A natural question to ask at this point is is there a difference between continuity and differentiability? In other words, can a function fail to be differentiable at a point where the function is continuous? To answer these questions, we consider a certain function \(f\text{,}\) where the graph of \(y=f(x)\) is displayed below in Figure1.98. We notice that \(f\) has a sharp corner at the point \((1,1)\text{,}\) and further observe that \(f\) is continuous at \(x=1\) since \(\lim_{x\to1} f(x)=1=f(1)\text{.}\)

Figure1.98A function \(f\) that is continuous at \(x = 1\) but not differentiable at \(x = 1\text{;}\) at right, we zoom in on the point \((1,1)\) in a magnified version of the box shown in the left-hand plot.

However, the function \(f\) in Figure1.98 is not differentiable at \(x = 1\) because \(f'(1)\) fails to exist. One way to see this is to observe that \(f'(x) = -1\) for every value of \(x\) that is less than 1, while \(f'(x) = +1\) for every value of \(x\) that is greater than 1. That makes it seem that either \(+1\) or \(-1\) would be equally good candidates for the value of the derivative at \(x = 1\text{.}\) Alternatively, we could use the limit definition of the derivative to attempt to compute \(f'(1)\text{,}\) and discover that the derivative does not exist. Finally, we can see visually in Figure1.98 that this function does not have a tangent line at \(x=1\text{.}\) Regardless of how closely we examine the function by zooming in on \((1,1)\) on the graph of \(y=f(x)\text{,}\) it will always look like a V and never like a single line, which tells us there is no possibility for a tangent line there.

SubsectionLocal Linearity

If a function does have a tangent line at a given point, then the function and the tangent line should appear essentially indistinguishable when we zoom in on the point of tangency.8For instance, see http://gvsu.edu/s/6J for an applet (due to David Austin, GVSU) where zooming in shows the increasing similarity between the tangent line and the curve. Conversely, if we zoom in on a point and the function looks like a single straight line, then the function should have a tangent line there and thus be differentiable at that point. Therefore, we say that a function that is differentiable at \(x = a\) is locally linear. An example of this can be seen in Figure1.99 below, with the graph of a function \(y=f(x)\) on the left and a magnified version with the tangent line to \(f\) at \(a\) on the right.

Figure1.99A function \(y = f(x)\) and its tangent line at the point \((a,f(a))\) is shown in the figure. Notice how the function resembles a line as we zoom in on \(x=a\text{.}\)
Example1.100

Let \(g\) be the function given by the rule \(g(x) = |x|\text{.}\)

  1. Visualize (or sketch) the graph of \(y=g(x)\text{,}\) and use it to explain why \(g\) is differentiable at every nonzero value of \(x\text{.}\)

  2. Use the limit definition of the derivative to show that \(g'(0) = \lim_{h \to 0} \frac{|h|}{h}\text{.}\) Then use left- and right- handed limits to explain why this limit does not exist. Conclude that \(g\) is not differentiable at \(x=0\text{.}\)

Hint
  1. What type of function is \(g\) for all \(x \lt 0\text{?}\) For all \(x > 0\text{?}\)

  2. Recall that \(g'(0) = \lim_{h \to 0} \frac{g(0 + h) - g(0)}{h}\text{.}\) What is the value of \(|h|\) when \(h \lt 0\text{?}\)

Answer
  1. \(g\) is piecewise linear.

  2. \begin{equation*} \lim_{h \to 0^-} \frac{|h|}{h}=-1, \ \text{and} \ \lim_{h\to0^+}\frac{|h|}h=1\text{.} \end{equation*}
Solution
  1. We know that \(g(x) = |x|\) is given by the formula \(g(x) = -x\) when \(x \lt 0\) and by \(g(x) = x\) when \(x \ge 0\text{.}\) Each of these pieces of \(g\) is a straight line, so at every point other than the point where they meet, the function \(g\) has a well-defined slope and thus is differentiable whenever \(x\neq0\text{.}\) The graph of \(y=g(x)\) is a translate of the earlier graph of \(y=f(x)\) from Figure1.98.

  2. Observe that

    \begin{align*} g'(0) =\mathstrut \amp \lim_{h \to 0} \frac{g(0+h)-g(0)}{h}\\ =\mathstrut \amp \lim_{h \to 0} \frac{|0+h|-|0|}{h}\\ =\mathstrut \amp \lim_{h \to 0} \frac{|h|}{h} \end{align*}

    To evaluate this limit, note that \(|h| = h\) whenever \(h > 0\text{,}\) and thus

    \begin{equation*} \lim_{h \to 0^+} \frac{|h|}{h} = \lim_{h \to 0^+} \frac{h}{h} = 1\text{.} \end{equation*}

    In contrast, we have \(|h| = -h\) whenever \(h \lt 0\text{,}\) and thus

    \begin{equation*} \lim_{h \to 0^-} \frac{|h|}{h} = \lim_{h \to 0^-} \frac{-h}{h} = -1\text{.} \end{equation*}

    Since the right- and left-hand limits are not equal, it follows that

    \begin{equation*} g'(0) = \lim_{h \to 0} \frac{|h|}{h} \end{equation*}

    does not exist. Therefore, the function \(g(x)=|x|\) is not differentiable at \(x=0\text{.}\)

SubsectionVertical Tangent Lines

Another example of when a function can fail to be differentiable at a point \(x=a\) is if the function has a vertical tangent at the point. In other words, when \(f\) is continuous at \(x=a\) and \(\lim_{x\to a}|f'(x)|=\infty\text{.}\) This means the tangent lines become very steep as we move closer to \(x=a\text{.}\)

Example1.101

In this example, let \(f(x)=\sqrt{x}\text{.}\)

In Figure1.102 below, we have the graph of \(y=f(x)\) along with a progression of tangent lines at points approaching \((0,0)\) on the graph. As we approach \(x=0\text{,}\) we see that the tangent lines drawn become steeper and steeper, ultimately leading to a vertical tangent line at \(x=0\text{.}\)

Figure1.102As we move closer to \(x=0\text{,}\) the tangent lines to the graph of \(y=f(x)\) become steeper and steeper. Notice that the tangent line closest to \(x=0\) is nearly vertical.

We can also show this by calculating the limit of the derivative close to \(x=0\text{:}\)

\begin{equation*} \lim_{x\to 0}f'(x)=\lim_{x\to 0}\frac{1}{2\sqrt{x}}=\infty. \end{equation*}

Therefore, \(f(x)\) is not differentiable at \(x=0 \text{.}\)

SubsectionLinks Between Continuity, Differentiability, and Limits

To summarize the preceding discussion of differentiability, we make several important observations.

  • If \(f\) is differentiable at \(x = a\text{,}\) then \(f\) is continuous at \(x = a\text{.}\) Equivalently, if\(f\) fails to be continuous at \(x = a\text{,}\) then \(f\) will not be differentiable at \(x = a\text{.}\)

  • A function can be continuous at a point without being differentiable there. In particular, a function \(f\) is not differentiable at \(x = a\) if the graph has a sharp corner (or cusp) at the point \((a,f(a))\text{.}\)

  • If \(f\) is differentiable at \(x = a\text{,}\) then \(f\) is locally linear at \(x = a\text{.}\) In other words, a differentiable function looks linear when viewed up close because it resembles its tangent line at any given point of differentiability.

Example1.103

In this example, let \(f\) be the function whose graph is given below in Figure1.104.

Figure1.104The graph of \(y = f(x)\) for Example1.103.
  1. State all values of \(a\) for which \(f\) is not continuous at \(x = a\text{.}\) For each, provide a reason for your conclusion.

  2. State all values of \(a\) for which \(f\) is not differentiable at \(x = a\text{.}\) For each, provide a reason for your conclusion.

  3. State all values of \(a\) for which \(f\) is not differentiable, but is continuous at \(x = a\text{.}\) Think about why this is the case.

Hint
  1. You might start by looking for places you would lift your pencil when drawing the graph, identifying points where \(f\) is not continuous. Calculating limits may also be useful.

  2. You might start by identifying points where \(f\) is not continuous.

  3. Thinking back to Example1.100 may be useful.

Answer
  1. At \(a = -2 \text{,}\) \(\lim_{x\to-2}f(x)\) does not exist; at \(a=-1\text{,}\) \(\lim_{x\to-1}f(x)\neq f(-1)\text{;}\) at \(a=2\text{,}\) \(\lim_{x\to2}f(x)\) does not exist; at \(a=3\text{,}\) \(f(3)\) is undefined.

  2. \(a = -2, -1, 2, 3\text{,}\) because \(f\) is not continuous at these points; \(a=-3,1\text{,}\) because \(f\) does not have a tangent line at these points.

  3. \(a = -3,1\text{.}\)

Solution
  1. \(f\) is not continuous at \(a = -2, 2\) because at each of these points \(\lim_{x\to a}f(x)\) does not exist. \(f\) is not continuous at \(a = -3\) because \(\lim_{x\to -1}f(x)=-3.5\text{,}\) but \(f(-1)=1\text{.}\) \(f\) is not continuous at \(a=3\) because \(f(3)\) is not defined.

  2. \(f\) is not differentiable at \(a = -2, -1, 2, 3\) because at each of these points \(f\) is not continuous. In addition, \(f\) is not differentiable at \(a = -3\) and \(a = 1\) because the graph of \(f\) has a corner point (or cusp) at each of these values.

  3. The only two points where \(f\) is continuous but not differentiable are \(a=-3,1\text{.}\) This is because of the corner point (or cusp). These points fit the criteria for continuity, but there is no discernible tangent line.

Example1.105

True or false: if a function \(p\) is differentiable at \(x = b\text{,}\) then \(\lim_{x \to b} p(x)\) must exist. Write at least one sentence to justify your choice.

Hint

What does being differentiable at a point tell you about continuity there?

Answer

True.

Solution

We know that a function \(f\) is continuous whenever it is differentiable, and that one characteristic of \(f\) being continuous at \(x=a\) is that \(\lim_{x\to a}f(x)\) exists. Therefore the statement if a function \(p\) is differentiable at \(x = b\text{,}\) then \(\lim_{x \to b} p(x)\) must exist is true.

SubsectionSummary

  • A function \(f\) is differentiable at \(x = a\) whenever \(f'(a)\) exists, which means that \(f\) has a tangent line at \((a,f(a))\) and thus \(f\) is locally linear at \(x = a\text{.}\) Informally, this means that the function looks like a line when viewed up close at \((a,f(a))\) and that there is not a corner point or cusp at \((a,f(a))\text{.}\)

  • Differentiability is a stronger condition than continuity, which is a stronger condition than having a limit. In particular, if \(f\) is differentiable at \(x = a\text{,}\) then \(f\) is also continuous at \(x = a\text{,}\) and if \(f\) is continuous at \(x = a\text{,}\) then \(f\) has a limit at \(x = a\text{.}\)

SubsectionExercises

Consider the graph of the function \(y = p(x)\) that is provided in Figure1.106. Assume that each portion of the graph of \(p\) is a straight line, as pictured.

Figure1.106At left, the piecewise linear function \(y = p(x)\text{.}\) At right, axes for plotting \(y = p'(x)\text{.}\)
  1. State all values of \(a\) for which \(\lim_{x \to a} p(x)\) does not exist.

  2. State all values of \(a\) for which \(p\) is not continuous at \(a\text{.}\)

  3. State all values of \(a\) for which \(p\) is not differentiable at \(x = a\text{.}\)

  4. On the axes provided in Figure1.106, sketch an accurate graph of \(y = p'(x)\text{.}\)

For each of the following prompts, give an example of a function that satisfies the stated criteria; a formula or a graph, with reasoning, is sufficient for each. If no such example is possible, explain why.

  1. A function \(f\) that is continuous at \(a = 2\) but not differentiable at \(a = 2\text{.}\)

  2. A function \(g\) that is differentiable at \(a = 3\) but does not have a limit at \(a=3\text{.}\)

  3. A function \(h\) that has a limit at \(a = -2\text{,}\) is defined at \(a = -2\text{,}\) but is not continuous at \(a = -2\text{.}\)

  4. A function \(p\) that satisfies all of the following:

    • \(p(-1) = 3\) and \(\lim_{x \to -1} p(x) = 2\)

    • \(p(0) = 1\) and \(p'(0) = 0\)

    • \(\lim_{x \to 1} p(x) = p(1)\) and \(p'(1)\) does not exist

Consider the function \(g(x) = \sqrt{|x|}\text{.}\)

  1. Use a graph to explain visually why \(g\) is not differentiable at \(x = 0\text{.}\)

  2. Use the limit definition of the derivative to show that

    \begin{equation*} g'(0) = \lim_{h \to 0} \frac{\sqrt{|h|}}{h}\text{.} \end{equation*}
  3. Investigate the value of \(g'(0)\) by estimating the limit in (b) using small positive and negative values of \(h\text{.}\) For instance, you might compute \(\frac{\sqrt{|-0.01|}}{0.01}\text{.}\) Be sure to use several different values of \(h\) (both positive and negative), including ones closer to 0 than 0.01. What do your results tell you about \(g'(0)\text{?}\)

  4. Use your graph in (a) to sketch an approximate graph of \(y = g'(x)\text{.}\)