Finding the second partial derivatives of a function is a useful skill to have. Second partial derivatives describe how a function changes with respect to two variables instead of one.
For example, let’s find the second partial derivative of the sine function with respect to x and y. Since the sine function only contains the x variable, we will need to find the derivative with respect to y.
To do this, we will first need to find the derivative of sine with respect to y, then subtract that value from 1. We will then need to find the derivative of sine with respect to x, then multiply them together. The reason we do this is because we are finding a derivative of a derivative!
This article will go into detail about how to find all second partial derivatives for any given funciton.
Multiply by 2 and find the second partial derivatives
A helpful technique is to multiply the function being differentiated by two, and then find the second partial derivatives. This can be done with any number of variables, and can even be done with unitized variables.
For example, let’s take a look at finding all the second partial derivatives of sine squared.
Check if they are equal to 0
Once you have found all of the second partial derivatives, you can go back to the original function and see if all of the second partial derivatives are zero.
If any of them are not zero, then your original function is not linear. A linear function is one where either its graph is a line, or every value of the output is calculated by only one value of the input.
For example, f(x) = 2x is a linear function because if you input any number x, then the output is always 2x. There is not a case where x = 2 and the output is 3x instead.
Repeat for all of the x and y coordinates
Once all of the partial derivatives have been found, you can find all of the second-order derivatives by repeating for all of the x and y coordinates.
First, find the derivative with respect to x for the function as a whole. Then, add in all of the contributions from x to find the total derivative with respect to x. Do the same for y.
For example, if F(x, Y) = Sin2(mx + Ny), then:
F’(x, y) = 2Sin2(mxy) + Cos2(my) + 2Cos2(mx + ny)
This is because:
Second-order derivatives are extremely useful in solving many different types of problems.
Put it all together to find F’(x, y)
To find the second partial derivative of a function, you need to find the derivative of the function and then add and subtract its derivatives of the opposite variable.
For example, let’s look at the following function: F(x, y) = Sin2(mx + Ny). We’ll call m the magnitude of x and n the magnitude of y.
To find F’(x, y), we need to find:
Second partial derivatives of x and y Second partial derivatives of -x and -y Second partial derivatives of sin2(mx + Ny) All these together!
Let’s start with finding all the second partial derivatives of x and y. These are going to be very similar to what we did before, except we will only do one side instead of doing both sides.
Use a computer to find F’(x, y)
Once you have found all of the second partial derivatives, you can find the general derivative by combining them all as follows:
F’(x, y) = F’(x, y) + F’(x, y) + F’(x, y)
This is an easy way to find the general derivative and is a good check that you have all of the second partial derivatives.
Once you have found the general derivative, you can use it to find any derivatives of functions that are not linearly related. For example, if f(x) = Sin2(mx + Ny), then f’(x) = 2Sin2(mx + Ny).
Using the General Derivative on Your Own Functions
You can use the general derivative on your own functions. To do this, first find all of the second partial derivatives of your function and then combine them all to get the general derivative.
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