On this screen we’re going to introduce differentials, a key Calculus concept, by building from the ideas you used in your simple calculations on the preceding screens. We’ll also use those ideas to lay the groundwork for how to determine the *rate* at which a function changes at a given point.

In the preceding Topic, we developed the method of **linear approximations** to compute a variety of values for a few different functions. For each calculation, the problem statement provided

- the function itself. For instance in Example 1, $f(x) = x^2.$
- a particular value of $x$ for which we can easily compute the function. For instance, at $x=3,$ $x^2=9.$
- the
*rate*at which the function changes at this particular value of $x.$ For instance, $\left.\dfrac{df}{dx}\right|_\text{at $x = 3$}=6$.

Given those pieces of information, we could then use our linear approximation method to compute the value of the function at a nearby value of

The table below shows each of the calculations we considered, starting with Example 1 in row 1, where $f(x)=x^2.$

Notice that in the second column we’re using the letter *a* to represent the *x*-value that we used as our “base point” from which we started our approximation. For instance, in row 1 we have $a=3$ and $f(a) = 3^2 = 9.$

Function | Value of $\pmb{a}$ | $\pmb{f(a)}$ | $\pmb{\left.\dfrac{df}{dx}\right|_\text{at $x = a$}}$ | We calculated the approximate value of: | Link to Problem in Preceding Topic |
---|---|---|---|---|---|

$f(x)=x^2$ | 3 | $3^2=9$ | $\left.\dfrac{df}{dx}\right|_\text{at $x = 3$}=6$ | $(3.01)^2 = (3+0.01)^2$ | Example 1 |

$g(x)=x^3$ | 1 | $1^3=1$ | $\left.\dfrac{df}{dx}\right|_\text{at $x = 1$}=3$ | $(0.99)^3 = (1-0.01)^3$ | Example 2 |

$f(x) = \sqrt{x}$ | 16 | $\sqrt{16}=4$ | $\left.\dfrac{df}{dx}\right|_\text{at $x = 16$}=0.125$ | $\sqrt{16.2} = \sqrt{16 + 0.2}$ | Practice Problem 1 |

$g(\theta) = \sin (\theta)$ | $0$ | $\sin (0) = 0$ | $\left.\dfrac{dg}{d \theta}\right|_\text{at $\theta = 0$}=1$ | $\sin(-0.13) = \sin(0-0.13)$ | Practice Problem 2 |

$g(\theta) = \sin (\theta)$ | $\dfrac{\pi}{3}$ | $\sin \left(\dfrac{\pi}{3} \right) = \dfrac{\sqrt{3}}{2}$ | $\left.\dfrac{dg}{d \theta}\right|_\text{at $\theta = \pi/3$}=\dfrac{1}{2}$ | $\sin(\pi/3 + 0.18)$ | Practice Problem 3 |

$g(\theta) = \sin (\theta)$ | $\pi$ | $\sin \left(\pi \right) = 0$ | $\left.\dfrac{dg}{d \theta}\right|_\text{at $\theta = \pi$}=-1$ | $\sin(\pi + 0.07)$ | Practice Problem 4 |

$g(\theta) = \sin (\theta)$ | $\dfrac{\pi}{2}$ | $\sin\left(\dfrac{\pi}{2}\right) = 1$ | $\left.\dfrac{dg}{d \theta}\right|_\text{at $\theta = \pi/2$}=0$ | $\sin\left(\dfrac{\pi}{2}+0.04\right)$ | Practice Problem 5 |

With that overview of calculations in mind, let’s summarize some key points about what we’ve done so far. While we’re using the particular computations we’ve completed to illustrate the following primary concepts, keep in mind that these ideas will apply to most functions we will encounter.

- In each problem, we focus on a point at $x=a$ (column 2) for which we know both (I) the function’s value $f(a)$ at that point (column 3), and (II) the
*rate*at which the function changes at that point, $\left.\dfrac{df}{dx}\right|_\text{at $x = a$}$ (column 4).

So far we’ve had to simply provide that rate for you; that’s about to change. - Using our linear approximation method we can calculate approximate values of the function for values of
*x*that are a small distance $dx$ away from $x=a.$ For instance, in Example 1 we had $dx=0.01.$ - We can envision our
*linear*approximation method as starting at the point we know about $\big(a, f(a)\big),$ and then walking along the*line*with slope equal to $\left.\dfrac{df}{dx}\right|_\text{at $x = a$}$ instead of following the function’s actual curve. - Walking along this line, the small change in the function’s output value $df$ is
**directly proportional**to the small change in*x*-value $dx$. The constant of proportionality is the function’s rate of change $\left.\dfrac{df}{dx}\right|_\text{at $x = a$}$:

\[df = \overbrace{\left( \left.\dfrac{df}{dx}\right|_\text{at $x = a$}\right)}^\text{rate of change at $x=a$} \cdot dx\] - The function’s rate of change $\left.\dfrac{df}{dx}\right|_\text{at $x = a$}$ is thus a measure of the function’s sensitivity at $x=a:$ the rate of change at that point determines how strongly the function reacts when you change its input by the small amount
*dx.*For example, in the the interactive figure below, at**(a)**$x=a_1$ the function’s rate of change $\left.\dfrac{df}{dx}\right|_\text{at $x = a_1$}$ is larger than at**(b)**$x=a_2.$ Using the slider beneath the graph you can vary the size of*dx*, which causes the identical changes in the horizontal direction in both (a) and (b). Observe that because $\left.\dfrac{df}{dx}\right|_\text{at $x = a$}$ is larger at $a_1$ than at $a_2,$ the function “reacts more strongly” at $a_1,$ and so the change*dy*is much larger at $a_1$ than at $a_2$ for the*same*change in*dx.*Graph of*f(x)*versus*x*

to illustrate effect of different $\left.\dfrac{df}{dx}\right|_\text{at $x = a$}$ values:Remember: $\color{blue}{df = \left(\left.\dfrac{df}{dx}\right|_\text{at $x = a$}\right) \cdot dx} $

Use the slider to vary the size of*dx*:

When you’re ready, check the box above to show points

**(c)**$x=a_3$ and**(d)**$x=a_4,$ where the function has negative rates of change. Notice that a positive value of*dx*produces a*negative*value for*df,*and the function’s value decreases.

The interactive graph below shows the function *g(x)* versus *x.* You can use the slider beneath the graph to vary the size of *dx* near each point. Choose the correct statement that orders the rates of change at the four points from most negative to most positive.

**(a)** $\left.\dfrac{dg}{dx}\right|_\text{at $x = a_1$} < \left.\dfrac{dg}{dx}\right|_\text{at $x = a_2$} < \left.\dfrac{dg}{dx}\right|_\text{at $x = a_3$} < \left.\dfrac{dg}{dx}\right|_\text{at $x = a_4$}$

**(b)** $\left.\dfrac{dg}{dx}\right|_\text{at $x = a_2$} < \left.\dfrac{dg}{dx}\right|_\text{at $x = a_4$} < \left.\dfrac{dg}{dx}\right|_\text{at $x = a_3$} < \left.\dfrac{dg}{dx}\right|_\text{at $x = a_1$}$

**(c)** $\left.\dfrac{dg}{dx}\right|_\text{at $x = a_2$} < \left.\dfrac{dg}{dx}\right|_\text{at $x = a_1$} < \left.\dfrac{dg}{dx}\right|_\text{at $x = a_4$} < \left.\dfrac{dg}{dx}\right|_\text{at $x = a_3$}$

**(d)** $\left.\dfrac{dg}{dx}\right|_\text{at $x = a_1$} < \left.\dfrac{dg}{dx}\right|_\text{at $x = a_3$} < \left.\dfrac{dg}{dx}\right|_\text{at $x = a_4$} < \left.\dfrac{dg}{dx}\right|_\text{at $x = a_2$}$

**(e)** $\left.\dfrac{dg}{dx}\right|_\text{at $x = a_4$} < \left.\dfrac{dg}{dx}\right|_\text{at $x = a_3$} < \left.\dfrac{dg}{dx}\right|_\text{at $x = a_2$} < \left.\dfrac{dg}{dx}\right|_\text{at $x = a_1$}$

Graph of *g(x)* versus *x* near different *x-*values

Use the slider to vary the size of *dx*:

Linear approximation means *direct proportionality* between small change in input and small change in output

You’ve probably noticed that in each of the graphs above, and in those accompanying each linear approximation calculation we’ve done, you see a \[df = \left(\left.\dfrac{df}{dx}\right|_\text{at $x = a$}\right) \cdot dx\]

Defining **differentials**

With that fundamental idea in mind, let’s introduce some new terminology: The small-change quantities - differentials are represented by placing a
*d*in front of the variable; - differentials represent
*small*changes in the variable’s value; - a function’s output-differential
*df*(or*dg*or …) changes at a*constant*rate with respect to the function’s input-variable’s differential*dx*(or $d\theta$ or …).

Leibniz notation and Leibniz triangles

Differentials were created by Gottfried Leibniz (1646-1716), which is why you might hear quantities like $\dfrac{df}{dx}$ referred to as

[Link to Wikipedia article about Leibniz.]

We wrote above (Point #1) that so far we’ve had to simply provide you with the *rate *at which a function changes at $x=a,$ and that that’s about to change. Indeed, a (the?) primary focus of this first part of learning Calculus is figuring out how to *determine* the value of $\left.\dfrac{df}{dx}\right|_\text{at $x = a$}.$ This is the same problem that the founders of Calculus faced.

So let’s shift our focus to this question, starting with a reversal of the problem-type we’ve been considering: in each problem we’ve examined so far, we’ve provided the function $f(x)$ (like $f(x) = x^2$), a point of interest, $x=a,$ and the value of $f(a)$ (like “at $x=3,$ $f(3) = 3^2 = 9),$” and the *rate* at which the function changes at that value of *x* (like $\left.\dfrac{df}{dx}\right|_\text{at $x = 3$}=6$). We’ve then essentially asked you to compute the value of *df* using

\[df = \left(\left.\dfrac{df}{dx}\right|_\text{at $x = a$}\right) \cdot dx\]
Let’s switch things up, and provide you with values for *df* and *dx* and then ask *you* to determine the rate $\left.\dfrac{df}{dx}\right|_\text{at $x = a$}.$ This is our first step toward finding multiple ways to determine the rate at which a function changes – or, said differently, toward determining the function’s exact “sensitivity” at the the point of interest.

To illustrate the idea, let’s return to that very problem of what happens when we vary the function $f(x)=x^2$ a bit, around the point $x=3.$ For the purposes of this Example, pretend that you do not know already know that $\left.\dfrac{df}{dx}\right|_\text{at $x = 3$}=6,$ and instead must determine that value.

The interactive graph below shows a zoomed-in version of the (by now familiar) graph of $f(x)=x^2$ near the point $x=3,$ along with the Leibniz triangle for this point and slider to vary the size of *dx* and hence *dy.*

If you’d like, you can zoom out to see that this really is part of the curve for $y=x^2.$ Then hit the “Home” button on the graph to return to the initial view.

**Step 1**. Zoom in even more and hide/show the red $y=x^2$ curve to (again) convince yourself that the hypotenuse of the Leibniz triangle closely mimics the function $f(x)=x^2$ curve’s behavior near $x=3.$ As usual, the smaller *dx* is, the better the green triangle’s line tracks the function’s red curve; if we extend *dx* to the ends of its range, we see how the green line deviates more and more from the red curve.

The key point here is to look and see for yourself that, “Yes, the triangle’s green line segment tracks the curve in this small region.” (Later you’ll have to put this green line in place yourself; here we’ve done that step for you.)

Once you’ve decided on that “yes” for yourself, please continue below.

Graph of $f(x) = x^2$ versus *x*

Use the slider to change the value of *dx*:

Currently*dx* = 0.010

Currently

(Remember that for the purposes of this Example, you do not yet know the value of $\left.\dfrac{df}{dx}\right|_\text{at $x = 3$}.$)

**Step 2.** We know that

\[df = \left(\left.\dfrac{df}{dx}\right|_\text{at $x = 3$}\right) \cdot dx\]
So using the interactive calculator, we can simply read off a set of values for *df* and *dx* in order to determine the constant $\left.\dfrac{df}{dx}\right|_\text{at $x = 3$}.$ For instance, if you set $dx=0.01$ you see $dy = 0.06$, and so we have

\[0.06 = \left(\left.\dfrac{df}{dx}\right|_\text{at $x = 3$}\right) \cdot (0.01)\]
and so we must have $\left.\dfrac{df}{dx}\right|_\text{at $x = 3$} = \; … ?$

Yes: 6!

You probably did that math in your head, but since the numbers won’t always work out so easily let’s solve for $\left.\dfrac{df}{dx}\right|_\text{at $x = 3$}$:

\[\left.\dfrac{df}{dx}\right|_\text{at $x = 3$} = \dfrac{0.06}{0.01}=6 \quad \quad \cmark\]

Hence we see that the Leibniz triangle itself – *if* we can get it aligned so that it looks to you like it mimics the function-curve’s behavior (for now, the only criterion we’ll use) – tells us the value of $\left.\dfrac{df}{dx}\right|_\text{at the point of interest $x=a$}.$

Just to double-check, you might choose a different value of *dx.* For instance, if you set $dx=0.005,$ you see $dy = 0.03,$ so

\[0.03 = \left(\left.\dfrac{df}{dx}\right|_\text{at $x = 3$}\right) \cdot (0.005)\]
and so again

\[\left.\dfrac{df}{dx}\right|_\text{at $x = 3$} = \dfrac{0.03}{0.005} = 6 \quad \checkmark \]

Time for you to try a problem, this time with a result you don’t already know.

The interactive calculator below shows a graph of some function $f(x)$ versus *x.* We're focused on the function's behavior around the highlighted point, $(4, 106).$
As before, once you have zoomed in sufficiently around this point, you will see the Leibniz triangle appear. Use the slider beneath the graph to vary the value of *dx,* and convince yourself that the triangle's line mimics the function's behavior near the point of interest. Then use the differential values *df* and *dx* to determine: $\left.\dfrac{df}{dx}\right|_\text{at $x = 4$} =$
\[ \begin{array}{lllll} \text{(A) }-2.4 && \text{(B) }2.4 && \text{(C) }-24 && \text{(D) }24 && \text{(E) none of these} \end{array} \]

Graph of *f(x)* versus *x*

Use the slider to change the value of *dx*: Currently *dx* = 0.1

**Differentials**are small changes in a variable’s value. For instance,*dx*is a small change in input value, and*df*is the resulting small change in the output value.- By definition,
*df*and*dx*are related according to $df = \left(\left.\dfrac{df}{dx}\right|_\text{at $x = a$}\right) \cdot dx,$ where $\left.\dfrac{df}{dx}\right|_\text{at $x = a$}$ is the*rate*at which the particular function changes at $x=a,$ and is thus a measure of the function’s “sensitivity” to changes in input-value at that point. - When we write this rate as $\dfrac{df}{dx},$ we are using
**Leibniz notation**, named after the person who invented it. The triangle that is formed with dx as its base and df as its height is known as a**Leibniz triangle**. - While on the preceding screens we provided the value of $\left.\dfrac{df}{dx}\right|_\text{at $x = a$},$ we are laying the groundwork for
*how*to determine it, which will lead to one of the Big Ideas in Calculus.

In the next Topic, we’ll take another big step toward your being able to determine this rate for yourself.

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