Why do we learn differentials?

In summary, differentials are a way to find the change of something based on the derivative of a function and some sort of variable like time or a distance or something. Differentials are used to find changes in things, like in our example of finding the change in y when x changes from 5 to 10.
  • #1
ShawnD
Science Advisor
718
2
One application of derivatives from first year calculus is something called differentials. The intent is to find the change of something based on the derivative of a function and some sort of varialbe like time or a distance or something.
Let's say you have this formula:
y = x^2
now here is the derivative:
dy/dx = 2x
now if you bring the dx over, it looks like this
dy = 2x dx

In math class, these are meant to find changes in things. Let's say you wanted to find the change in y when x changes from 5 to 10. you would just fill in the equation like this:
dy = 2(5)(5)
dy = 50

the dy is your change in y. the first 5 is your original x value. the second 5 is your change in x.

The differential said the change is 50. Now let's see what the original equation says the difference is:
final - original
= x^2 - x^2
= 10^2 - 5^2
= 100 - 25
= 75

The two different equations give VERY different answers. They're not even close. Knowing this, why do we still learn these?
 
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  • #2
dy/dx is the instantaneous rate of change of y with respect to x. Since y(x) = x2 is nonlinear, you can't expect dy/dx to equal Δy/Δx
 
  • #3
Exactly. If they don't work then why the hell do we learn them?
 
  • #4
Because differentials work well then the quantities involved are small.

Let's stick to the [itex]f(x)=x^2[/itex] example, but with a smaller differential... how about [itex]x=10[/itex] and [itex]\delta x=1[/itex].

In this case, we have [itex]f(11)=121[/itex] and the differential approximation gives [itex]f(11) \approx f(10) + 1 * f'(10) = 120[/itex] which is pretty darn close.

The relevant theorem is:

[tex]
f(x + \delta x) = f(x) + f'(x) \delta x + \varepsilon (\delta x) \delta x \
\mathrm{where} \lim_{\delta x \rightarrow 0} \varepsilon(\delta x) = 0
[/tex]

In other words, the error term in the approximation of [itex]f(x+\delta x)[/itex] shrinks "quickly" as [itex]\delta x[/itex] approaches 0. In fact, if [itex]f(x)[/itex] is twice differentiable, you can prove that there exists a constant [itex]c[/itex] such that [itex]|\varepsilon (\delta x)| < c |\delta x|[/itex], so the error term is quadratic in [itex]\delta x[/itex]. (This is the Taylor remainder theorem; a differential approximation is just a first degree Taylor polynomial!)


edit: finally got the LaTeX right. :smile:
 
Last edited:
  • #5
oh ok, that makes sense.
 

1. Why is it important to learn differentials?

Differentials are an essential tool in mathematics and science, particularly in the fields of calculus and physics. They allow us to understand and analyze the rate of change of a function, which is crucial in modeling and predicting real-world phenomena. Additionally, differentials are used to solve complex equations and optimize functions, making them a valuable tool in problem-solving.

2. How are differentials used in real-world applications?

Differentials have countless real-world applications, from predicting the trajectory of a projectile in physics to optimizing the production process in economics. They are also used in engineering, biology, and many other fields to understand and model the behavior of complex systems. In finance, differentials are used to calculate interest rates and make investment decisions.

3. Can you explain the concept of a differential in simple terms?

A differential is the rate of change of a function at a specific point. It represents the instantaneous change in the value of a function as the input variable changes slightly. Think of it as the speedometer in a car - it tells you how fast you are going at a particular moment, rather than your average speed over a distance. In mathematics, the differential is represented by the symbol "d" followed by the variable of the function.

4. How does learning differentials benefit us in our everyday lives?

While differentials may not seem directly applicable to our daily lives, their concepts and techniques are used in various fields such as economics, medicine, and engineering. Understanding differentials can help us make informed decisions, analyze data, and solve problems in our personal and professional lives.

5. Are there any common misconceptions about differentials?

One common misconception about differentials is that they are only applicable in advanced mathematics and science. In reality, differentials are used in various fields and can be understood by anyone with a basic understanding of calculus. Another misconception is that differentials are only used to calculate rates of change. While this is one of their primary purposes, they also have other important applications, such as optimization and approximation.

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