00:01
In this question, we are asked to calculate some of the second order partial derivatives of the given function.
00:06
But first, we need to calculate the first order partial derivatives.
00:12
To calculate the derivative with respect to x, we can assume that y is a constant.
00:18
Since y is a constant, we can factor out 6 cos y to the 8.
00:29
Then, the derivative of x to the 7th equals to 7x to the 6, equals to 7x to the 6.
00:40
And then, we'll get 42x to the 6 multiplied by cos y to the 8th power.
00:48
So, this is the first derivative.
00:52
Now, let's calculate fxy.
00:55
Fxy is simply the derivative of fx with respect to y.
01:00
So, it's d over dy of 42x to the 6 times cos y to the 8th.
01:07
Since we are differentiating with respect to y, we can assume that x is a constant.
01:14
So, we can factor out 42x to the 6.
01:19
And then, to calculate the derivative of cos y to the 8th, we'll use the chain rule.
01:25
By the chain rule, we need to differentiate cosine first.
01:28
And the derivative of cosine is negative sine y to the 8th.
01:34
And then, multiply by the derivative of y to the 8th, which is 8y to the 7th.
01:39
Now, let's simplify this.
01:42
We'll get negative 42 times 8 equals to 336 times x to the 6 y to the 7 times sine y to the 8th power.
01:56
So, this is fxy...