Calculus is the story about the very small. We talk about quantities that are smaller
than any specific number. “Infinitesimal” is the mathematics word for these
quantities. Since infinitesimals are smaller than any of our numbers, we can’t describe
them with our numbers.
The language of limits is our way of discussing the concepts of infinitesimals.
For rates of change, the infinitesimal story involves instantaneous rate of change. Our tool for measuring this type of change comparison is called the derivative.
We have a couple of ways of representing the derivative of a function.
\(\blacktriangleright \) Prime Notation
Suppose \(f\) is the name of a function.
The derivative of \(f\) is a new function and it is denoted as \(f'\).
The little “tick mark” in the exponent position is called the prime sign.
\(f'\) is pronounced as “f prime”.
The derivative measures the change in the function values compared to changes in the domain values - at the infinitesimal level. It may not seem like it now, but, this is (will be) straightforward for most situations. However, we often encounter situations where our function representation involves parameters, like
Which symbol is representing the domain?
Or, more to the point, functions of more than one variable:
To which variable is \(f'\) referring?
It can be confusing at times. Therefore, we have an alternative notation for derivatives to clear up the confusion.
\(\blacktriangleright \) Leibniz Notation
Gottfried Leibniz was a German mathematician who was investigating the concepts of Calculus at the same time as Newton. While Newton developed his “dot” notation for derivative, Leibniz invented a more useful notation.
The derivative of \(f\) with respect to \(x\) is represented as \(\frac {df}{dx}\).
\(\frac {df}{dx}\) is pronounced as “dfdx”. It looks like a fraction, but it isn’t.
“with respect to \(x\)” means \(x\) is representing the domain for rate measurment purposes.
Just like with other mathematical notation, we allow useful modifications to this notation.
As the symbol \(\frac {df}{dx}\) shows, the function should be on top next to the \(d\). We should write \(\frac {d sin(x)}{dx}\). However, if the formula for the function gets long, then it becomes difficult to read the notation. For instance,
Therefore, it is sometimes clearer if the function is placed to the right of the differentiation notation:
Calculus will dive deep into all of this notation.
We are just introducing it and bringing it into our vocabulary, so that we can recognize and follow the symbols in Calculus, because it tends to go by fast there.
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more examples can be found by following this link
More Examples of Rates of Change