Tag: differentiation

After learning the rules (power rule, product and quotient rules, chain rule, etc.) and doing numerous exercises, a student realizes that the mechanics of Differential Calculus are not that difficult. Actually, they are almost boring, if it wasn’t for that one rule learned after all the other rules: implicit differentiation. ...

I am currently learning about this very powerful calculus tool. This tool is of great value in simplifying some functions prior to differentiation. I will try to explain simply in my own words. What are logarithms? Logarithms were invented by John Napier (1550-1617) for the purpose of simplifying calculation; basically ...

A few days ago was I wrote about Linear Approximations. These are used, as the name implies, to approximate the value of a function, usually a complicated function, by way of using a much simpler linear function, as long as \(\boldsymbol{\Delta x}\) (the change in x) is small. Along with ...

I have been trying to learn Calculus on and off for several years. In the last 3 months I re-read about Limits, Differentiation and Applications of Differentiation and finally, reached the point where I am starting to learn about Integration. It occurred to me that it might be interesting to ...

Every Algebra student knows the distance formula; that is, how to calculate the distance between two points on the Cartesian coordinate plane: \(\boldsymbol{d = \sqrt{\left(x_{2} – x_{1}\right)^{2} + \left(y_{2} – y_{1}\right)^{2}}}\). This week’s blogpost is about finding the distance between a specific point and the closest point(s) in a function. ...

One of the many interesting things I’ve come across while learning calculus is the concept of concavity. A simple example of a concave function is a parabola. The left graph below shows a concave up function: \(\boldsymbol{y = x^{2}}\), while the right graph shows a concave down function: \(\boldsymbol{y = ...

Recently I wrote about using the First Derivative (FD) Test to find the intervals where a function is increasing or decreasing. With the FD Test, we can also find the exact value where the function is highest or lowest in an interval (called absolute maxima) or, where it is higher ...

Sometimes it is easy to see exactly where a function has its highest or lowest values just by looking at its graph. Most often, though, the answer requires analytical work. I am starting this new Blog with an example on how to find analytically on what interval(s) a function is increasing ...