Gerald has taught engineering, math and science and has a doctorate in electrical engineering.
Review of Partial Derivatives
Fred has mastered differentiating when a function has only one variable. Somehow, he missed the lessons on partial differentiation. He's starting to panic because the current homework deals with higher-order partial derivatives.
In this lesson, we review partial differentiation by using examples. Then, we extend the method to higher-order partial derivatives. Let's zoom in on Fred to see what he's currently trying to figure out.
Our fearless math student, Fred, is being asked to find the derivative of x4 - x3. He correctly answers: 4x3- 3x2. Let's help Fred with his review of partial derivatives. You ask him to find the derivative of x4a - x3b where a and b are constants. Again, a correct answer: 4x3a- 3x2b.
Now the fun begins. Let's read the following math statement:
It says: the partial (derivative) with respect to x. The word ''derivative'' is in parentheses because we often just say: the partial with respect to x.
Let's write the derivative of x4a - x3b as a partial derivative:
The answer is still 4x3a- 3x2b.
The a and the b are constants. When taking the partial derivative with respect to x, everything other than x is a constant. So we could write:
and y6 and y5 are treated as constants. So, the answer is still the same except the a and b constants are more specific. The answer is now x4y6 - x3y5. Thus, if f(x, y) is the function:
and y6 and y5 are treated as constants
Of course, we can also have a partial with respect to y where everything else is constant. To do these kinds of derivatives, you develop a selective focus with your eyes. You ''see'' the y as the variable and the x4 and the x3 as constants.
And this is pretty much the idea with partial derivatives. You just have to be careful with the ''with respect to'' part. Fred is ready to move on to higher-order derivatives.
Let's say we want to take the partial derivative of the partial derivative. We write this as:
and read this as the second partial with respect to x. This is called a second-order partial derivative. Using the example for f(x, y), here are the details:
- Line 1: The partial squared with respect to x squared of f with respect to x squared of f is the partial with respect to x of the partial of f with respect to x. We have already computed the quantity in parentheses.
- Line 2: Replace the quantity in parentheses with 4x3y6 - 3x2y5.
- Line 3: Take the derivative with respect to x gives 12x2y6 - 6xy5.
For practice, we ask Fred to work on the second partial with respect to y of f. The details:
This idea of the partial derivative of the partial derivative of the … can be extended to even higher-order derivatives.
Fred is overjoyed at his success with higher-order partial derivatives. But wait, there's one more item to check out.
Mixed Partial Derivatives
We can also mix the partial derivatives. Mixed partial derivatives, in the case of two variables, are where the partial derivative is taken with respect to one variable to get a result. The first time we differentiate, we take the derivative with respect to y. Then, we differentiate again but with respect to x. Using the same f(x, y) as before:
Fred very astutely asks if the order matters. He means first differentiate with respect to x and then with respect to y:
If the function is continuous over x and y, then the mixed partial will be the same using either order. Fred is completely content and ready to try the quiz questions. But first, some more examples.
Find the third partial with respect to z of f(x, z) = ez + x cos z.
First, let's take the partial derivative of f with respect to z:
Do you recall? As we can see, this partial with respect to z is now ez + x cos z.
- The derivative of ez = ez
- The derivative of cos z = - sin z
- The derivative of sin z = cos z
Then, compute the partial squared with respect to z2:
For f(x, y, z) = 2x2yz3, find the mixed partial derivative of f with respect to x and z.
We are looking to calculate:
Let's start with the partial with respect to z:
Then, take the partial derivative with respect to x, which is:
And we're done.
When differentiating a function with more than one variable, we can differentiate with respect to one variable while keeping all the other variables constant. If x is the variable to be differentiated, we call this the partial with respect to x. A second-order partial derivative involves differentiating a second time. In both the first and second times, the same variable of differentiation is used. This idea may be extended to even higher-order partial derivatives.
Another type of higher-order partial derivative is the mixed partial derivative where, in the case of two variables, the partial derivative is taken with respect to one variable to get a result. Then, the partial derivative of the result is taken but with respect to the other variable. When the function is continuous, changing the order of the variables of differentiation will not change the final result.
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