Hi there. In this mathematics post, I cover the basics of partial derivatives. It is assumed that the reader is familiar with calculus derivatives for a single variable.
Quicklatex.com is used for rendering math text and symbols into images. I do not cover product rule, quotient rule nor Chain rule for partial derivatives here.
Topics
- Some Basic Calculus Derivatives
- From One Variable To Two Variables
- Partial Derivatives Basic Examples
Some Basic Calculus Derivatives
Calculus derivatives help with find the slope of a mathematical function at any point x. For geometric illustration, you can draw a tangent line at the point (a, f(a)). The slope of the tangent line would be equal to the derivative of the function at the point a.
From https://i.ytimg.com/vi/0rbuzOHup74/maxresdefault.jpg
Some Basic Derivatives
Product Rule
The product rule in calculus is for cases where there is a product of two different types of functions. Different types of functions include polynomials, exponential functions, trigonometric functions, square root functions, logarithmic functions. Here is the product rule formula given two functions f(x) and g(x).
Personally, I like the shorthand notation. This assumes that x is the independent variable.
Here is one product rule example.
Quotient Rule
For dealing with two functions with division, there is the quotient rule for the derivative for a quotient of different functions.
From One Variable To Two Variables
The one variable case for derivatives is a special case of multivariable calculus. It is sometimes not realistic to deal with one variable that affects the dependent variable. There may be other variables that affect the value of a dependent variable. This is why multivariable calculus exists.
The notation is slightly different with multivariable calculus and there are more variables to deal with. For the partial derivative with respect to x, you can have these notations. With respect to x means that the independent variable is x and the variable y is treated like a constant/number.
Second Partial Derivatives
Reference: https://tutorial.math.lamar.edu/classes/calciii/highorderpartialderivs.aspx
The notation for second derivatives are as follows. As there are two variables, you have different orderings in taking derivatives. You can take the first derivative with respect to x or y and the second derivative is with respect to x or y. There are four different derivatives.
Note that the notation here is not easy on the eyes. One of the challenges with math is reading notation. A bit of an eye test.
Partial Derivatives Basic Examples
This section covers basic examples of partial derivatives.
Example One
In the first example, I cover a simple multivariable function.
Example Two
Consider z = g(x, y) = x^2sin(y).
Example Three
For this last example I mix an exponential function with a logarithmic function. Note that the derivative of e to the x is still e to the x. I do skip some steps here.
I'll have to read those other topics to get into this.
Nice post all in all.
I might use that site you posted.
This topic does require knowledge of introductory calculus (single variable). You may want to consider https://math.typeit.org/ to help with typing some math symbols.
Cool. Thank you so much
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