Differentials

Engineering Context:

An understanding of differentials informs our understanding of the basics of calculus. Understanding differentials also helps us understand rates of change, an essential concept for engineers designing for dynamic systems. Although differentials seem to be a fairly theoretical concept on their own, they are an essential foundation of more advanced, highly applicable topics, such as calculus and differential equations. Differentials are also helpful when estimating values that are hard to calculate directly.

The Essentials

Differentials are, essentially, very small changes in input or output of a function. In calculus, they are typically considered to be infinitesimals (infinitely small in value). They are notated as da, with a being any letter (dy, dx, ds, etc). Conventionally, dx is the differential for a change in input and dy the one for a change in output.

dx can be thought of as an infinitely small ∆x and dy as an infinitely small ∆y. This association informs the use of differentials in the notation of a derivative, $\frac{dy}{dx}$ , as shown below:

Figure 1: The graph $f\left(\mathrm{x}\right)=\frac{1}{2}{x}^2$ , scaled normally on the left and zoomed in to the point (1, .05) on the right. As you zoom further in to this specific point and look at increasingly small values of ∆y and ∆x, those values will eventually become infinitely small differentials dy and dx, and the slope between them the derivative $\frac{dy}{dx}$ at the point (1, 0.5).

Differentials provide a method for estimating how much a function changes as a result of a small change in input. If $\frac{dy}{dx}=f\prime \left(x\right)$ , this equation can be rearranged to the differential form $dy=f\prime \left(x\right)dx$ , which is similar to a linear equation $y=mx+b$ ( $f\prime \left(x\right)$ is comparable to the slope m). Using the differential equation form, you can find a linear approximation of a function at a given point:

which is the approximation for the function’s change in input when x changes by 0.1. The actual input change between x = 3 and x = 3.1 is:

which is very close to our approximated value of .9. Essentially what this equation is doing is taking the instantaneous slope at our initial point $x=3$ and assuming that that slope will be approximately constant across a small interval $dx=.1$ , from $x=3$ to $x=3.1$ .

Differentials can also be used to estimate error. If the actual value of some measured quantity is a but the measured quantity is $a+dx$ , the error in any function of this quantity is $\mathrm{∆y}=f\left(a+dx\right)-f\left(a\right)$ , which can be approximated by $dy=f\prime \left(a+dx\right)dx$ .

A Deeper Dive

You can think of a derivative as a local analysis of the relationship between input and output around a certain point. In earlier math classes, we used $\frac{\mathrm{rise}}{\mathrm{run}}$ to find the slope a function. Finding a derivative uses essentially the same process, but using infinitely small differential values for rise and run. Since slope of a function is given by $\frac{\mathrm{∆y}}{\mathrm{∆x}}$ or $\frac{\mathrm{rise}}{\mathrm{run}}$ and since a derivative is thought of as an instantaneous slope, it would make sense that a derivative $\frac{\mathrm{dy}}{\mathrm{dx}}$ is the slope of an infinitely small interval, with an infinitely small change in y, dy, over an infinitely small change in x, dx.

Differentials are also used in the notation of an integral:

Here, the use of the differential dx emphasizes that an integral is an infinite sum of infinitely small quantities. We can see the significance of this notation when looking at the transition from Riemann sums to integrals. A Riemann sum approximates the area under a curve using the formula

on a closed interval $\left[x_a,x_b\right]$ . n refers to the number of subintervals the interval is divided into, i is the designation of each subinterval, $x_i^{\ast }$ a point anywhere on each subinterval, and ∆x is the width of each subinterval.

To get from a Riemann sum to an integral, take the limit of the sum as n goes to infinity:

As n, the number of subintervals, goes to infinity, ∆x will get smaller and smaller in value until it is the infinitely small differential dx. It is important that this differential value stay in the notation of an integral so we know which variable we are integrating with respect to – in this case, that we are taking the infinite sum of a function by summing infinitely small partitions of x. This notation comes in particularly handy when performing variable substitutions to compute integrals.

Practice

Exercise 1. If $f\left(x\right)=x^2+4x+2$ and $dx=.2$ , find dy at $x=5$ .

Exercise 2. If a cube is measured to have 5.0 inch sides with a possible measurement error of .1 inches, what is the estimated error of the volume of the cube?

Solution 1. Using the equation $dy=f\prime \left(x\right)dx$ , $dy=2.8$ . This is a good approximation for how much the function changes between x = 5 and x = 5.2.

Solution 2. Use the equation $dy=f\prime \left(a+dx\right)dx$ , where $f\left(x\right)$ is the formula to calculate volume of a cube ( $x^3$ ), a is the side length of the cube (5), dx is error in initial measurement (.1), and dy = is estimated volume error. We will come out with dy = 7.803. This is approximately how much error a calculation for the volume of the cube will have if each side length has a full .1 inch error.

More Resources

• Organic Chemistry Tutor: Differentials and Derivatives External Link Icon
• Lamar: Intro to Differentials External Link Icon