Calculus - Product Rule
Hello. Here is an overview of the product rule from calculus.
It is assumed that the reader knows what a calculus derivative is and is familiar with derivatives of functions such as 
, 
and so on.
(Math text, symbols are done in LaTeX and converted into images with QuickLaTeX.com)
The Product Rule
The product rule is used when you want to take the derivative of one function f(x) multiplied by another (different) g(x) function. We have:
The derivative of h(x) is h'(x) and the form is as follows:
Think of the derivative h'(x) in as the derivative of the first multiplied by the second function + first function multiplied by the derivative of the second function.
A Memory Aid For Product Rule
There is a neat memory aid for the calculus product rule. Instead of using the argument x in f(x), I use just f and g from g(x).
Instead of , an easier way is to use 
.
Examples of Product Rule
Example One
Suppose that 
and h(x) is in the form of 
.
Here we have f(x) = x with f'(x) = 1.
and 
with 
.
Substituting the components into the product rule formula would give h'(x) as:
Example Two
What is the derivative of 
The product rule formula would give h'(x) as:
Example Three
Determine the derivative of 
.
With the product rule formula, this would give:
We do this at the first year at university bachelor degree during the first semester. I love it and still remember something about limits and derivatives. Studying how a function behaves and finding minimum and maximum was really great work. I like the integrals as well. But during second semester we would go further with double and even triple integrals and that was a really hard part to understand :-) Partial derivatives concept was also not very easy to grasp.
Calculus can be fun at times.
It depends on the region where and when calculus is taught. In the province of Ontario in Canada, Calculus is offered in grade twelve (age 17/18). Most universities here offer a standard calculus course while places like University of Toronto/Waterloo offer calculus more of a real analysis type course.
My multivariable calculus knowledge is a bit rusty. I do remember Fubini's Theorem sort of.
Pure mathematics scares
hahaha...true but not all the time...only during finals😂