00:01
For this problem, we are asked to compute all second derivatives for the function f of xy equals 1 over co -squared x plus e to the power of negative y.
00:09
We'll start by taking the partial derivative of f with respect to x.
00:14
Now, we'll do this by applying, or by first treating y as a constant.
00:20
Then we'll be applying the chain rule.
00:22
So we would take the derivative of 1 over co -squared x plus e to power of negative y with respect to the denominator, which will first give us negative 1 over co -squared x plus e to the power of negative y squared times the partial derivative of co -squared x plus e to power of negative y with respect to x.
00:45
So that would be multiplied by negative sign of 2x.
00:48
So we can see that the partial derivative then with respect to x will be sign of 2x over co -squared x plus e to the power of negative y all squared.
01:02
We'll take the second partial derivative of this with respect to x.
01:08
Now taking the second partial derivative, we'll need to apply the quotient rule.
01:12
In the expanded form, we'll have that this will become cos of 2x times 2, cos squared x, plus e to the power of negative y, squared minus negative 2, sine of 2x, of 2x times co -squared x plus e to the power of negative y e to power of negative y times sign of 2x all divided by co -squared uh oops co -squared of x plus e to the power of negative y to the power of 4 which means that our second partial derivative with respect to x will be 2 times sine squared of 2x plus cos of 2x times co squared of x plus e to the power of negative y divided by co squared x plus e to the power of negative y all cubed now we'll proceed by taking the partial derivative with respect to y which will give us that the first derivative with respect to y going through the same procedure holding x as a constant we'll find that the first derivative will be e to of negative y over co squared of x plus e to power of negative y squared and we'll find that the second partial derivative with respect to y will be equal to e to the power of negative 2 y times e to the power of y, co -squared of x plus 1, divided by co -squared of x plus e to the power of negative y, all cubed.
03:06
Now we'll take a look at finding our mixed partial derivatives...