Find all the second partial derivatives. v = e^9xe^y vxx =...

Find all the second partial derivatives. v = e^{9xe^y} v_{xx} = 81e^{2y}e^{9xe^y} v_{xy} = 9e^y e^{9xe^y} 9xe^y + 9e^y e^{9xe^y} v_{yx} = 9xe^y e^{9xe^y} v_{yy} = 9e^y e^{9xe^y} + 9xe^y e^{9xe^y} 9e^y

Transcript

00:01 So we are given v is equal to e to the power of 9x e to the power of y.

00:08 So we need to find out all the second partial derivatives.

00:11 So let's start with vx first.

00:14 So vx means partially differentiating with respect to x.

00:18 So differentiation of e power x is the same which is 9x e power of y.

00:22 Now you will be differentiating this function that is x into e power of y where y will be treated as constant.

00:29 So 9 e power y will be written as it is and the differentiation of x will be 1.

00:35 So thus the value of vx has come out to be 9 e to the power of y in multiplication with e to the power of 9x e power of y.

00:48 Now let's differentiate with respect to x again.

00:51 So vxx over here will be equals to because y is constant.

00:55 So 9 e power of y will be written as it is outside the bracket and then you can just differentiate this e power of 9x e power y again.

01:04 So e power 9x e power of y is the same.

01:07 Differentiation of 9x e power y again y will be treated as constant.

01:12 9 is constant e power y is constant and differentiation of x with respect to x will be 1.

01:18 So thus you will be left with vxx which is equal to 9 times 9 is 81 and this is 81 e power y same base.

01:27 So the powers are added.

01:28 So this is 2y e power of 9x e power of y.

01:33 So this is vxx.

01:34 Now let's just try to differentiate equation 1 with respect to y.

01:39 So x will be treated as constant.

01:41 So this is 9x e power of y.

01:44 So if x is written as constant, so 9x will be written as it is and the differentiation of e power y will be e power y.

01:51 So you get the value of vy as 9x e power of y e power of 9x e power of y.

02:00 Now let's differentiate this equation again with respect to y.

02:04 X is constant.

02:05 So 9x is written as it is and in these two functions of x treat e power y as the first function.

02:12 So e power y is the first function differentiation of e power of 9x e power of y will be the same and x will be treated as constant.

02:21 So 9x is constant and differentiation of e power y is e power y.

02:25 Then you write plus of because now you're taking e power of 9x e power y.

02:30 That is the second function differentiation of the first.

02:33 So e power y is e power y.

02:35 So thus you can simply write vyy as...

Question

Find all the second partial derivatives. v = e^{9xe^y} v_{xx} = 81e^{2y}e^{9xe^y} v_{xy} = 9e^y e^{9xe^y} 9xe^y + 9e^y e^{9xe^y} v_{yx} = 9xe^y e^{9xe^y} v_{yy} = 9e^y e^{9xe^y} + 9xe^y e^{9xe^y} 9e^y

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