Lecture Note
University
Massachusetts Institute of TechnologyCourse
Multivariable CalculusPages
3
Academic year
2023
Sporkz
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98
Higher order partial derivatives We will first compute the partial derivative of xy squared plus x squared y withrespect to x, then we will compute its derivative with respect to y, and finally weevaluate the partial derivative in the x direction at (1, 2) to the first problem. In thesecond problem, we will compute second partial derivatives by taking the derivativeof derivatives, as we did in one-variable calculus. To find the derivative of xy, we multiply by y. The result is f(x).In order to find itspartial derivative in the x direction, we treat y as a constant and take an ordinaryderivative with respect to x. We then find that this partial derivative equals y squared.Since we differentiate only x and not y, the derivative of x squared is 2x.
For the partial derivative in the y direction, we do the same thing except now x is aconstant and we are taking an ordinary derivative in the y direction. So 2xy plus xsquared. Finally, we evaluate the partial derivative of f(x) with respect to x at (1, 2).This means plugging in x equals 1 and y equals 2 into our previous computation. Sowe get 2 squared plus 2 times 1 times 2, which equals 8. This is how we computepartial derivatives. Now we will compute the second partial derivatives of our function. In other words,we will work out how to find the partial derivative of our function when x varies byone unit in either direction. For instance, if we want to take the partial derivative of (y)with respect to x, again in the x direction, all this means is that when we took the firstpartial derivative of (y) with respect to x, we got a function of x and y. And now weneed only take its partial derivative again in the x direction. The derivative of y squared, be careful the derivative of y squared in the x direction isjust 0, because y is a constant, relative to x, and we just get 2 y. When we take thederivative of this x, we just get 1. So that's our partial derivative in the x direction.And now, we can also take mixed partials. So here, we take a derivative of f. First, we take the derivative in the y direction andthen take a derivative in the x direction. The result is that we have partial f, partial y,which we need to take in the x direction, and so we get 2y plus 2x. Now, let's see what happens when we switch the order of partial derivatives and takea derivative of the partial derivative in the opposite direction. So now let's go back toour partial derivative in the x direction and take its derivative in the y direction. Thefirst term there gives us 2y and the second term gives us 2x. It should be noted that the mixed partial derivatives of polynomial and differentiablefunctions in xy and yx orders are equal.
There are many examples of polynomial functions where the denominator andnumerator are not equal. However, certainly for any rational function, thesequantities will always be equal.
Higher Order Partial Derivatives
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