We are given the function f of x coma y,
and asked to find the second order partials.
Before we can find the second order partials,
we need to find the first order partials.
Let's begin by determining the partial of f
with respect to x,
which means we will differentiate f of x coma y
with respect to x, treating y as a constant.
Which means you will treat cosine y to the ninth
as a constant.
And since the derivative of 4 x to the fifth
with respect to x is 20 x to the fourth,
the partial f with respect to x
is 20 x to the fourth times cosine y to the ninth.
And now let's find the partial of f with respect to y,
differentiating f of x coma y with respect to y,
now treating x as a constant.
And therefore we treat 4x to the fifth as a constant,
and then multiply by the derivative of cosine y to the ninth
with respect to y, which is equal to
negative sine y to the ninth times 9 y to the eighth
or negative 9 y to the eighth times sine y to the ninth.
Notice how we had to apply the chain rule
to differentiate cosine y to the ninth with respect to y.
Let's go ahead and re-write the partial of f
with respect to y,
as a negative 36 x to the fifth
y to the eighth sine y to the ninth.
And now we will use the first order partials
to find the second order partials.
Let's first find the second order partial with respect to x
then with respect to y,
which means we will differentiate the partial of f
with respect to x, with respect to y,
now treating x as a constant.
So if we differentiate 20 x to the fourth
cosine y to the ninth with respect to y now,
treating x as a constant.
We will have 20 x to the fourth
times the derivative of cosine y to the ninth
with respect to y,
which is equal to negative sine y to the ninth
times 9 y to the eighth
or negative 9 y to the eighth times sine y to the ninth.
Let's go and re-write this as negative 180 x to the fourth
y to the eighth sine y to the ninth,
let's also record this above.
And now let's find the second order partial
with respect to y,
which means you will now differentiate
the first order partial with respect to y
with respect to y again.
So we will differentiate negative 36 x to the fifth
y to the eighth sine y to the ninth with respect to y,
treating x as a constant.
And therefore we'll have negative 36 x to the fifth
times the derivative of y to the eighth sine y to the ninth
with respect to y,
so I have to apply the product rule
where we have the first function of y to the eighth
times the derivative of the second function,
which is a derivative of sine y the ninth,
which is equal to cosine y to the ninth
times 9 y to the eighth
or 9 y to the eighth times cosine y to the ninth
and then we have plus the second function of sine
y to the ninth times the derivative of the first function
with respect to y,
which is the derivative of y to the eighth with respect to y
which is 8 y to the seventh.
Let's go and distribute the negative 36 x to the fifth,
negative 36 times 9 is negative 324,
so you have negative 324 x to the fifth
y to the sixteenth cosine y to the ninth,
and then we have plus negative 36 times eight
is negative 288,
so let's write that as minus 288 x to the fifth
y to the seventh sine y to the ninth.
The other option would be to factor this,
but I'm going to go and leave it in this form.
So let's go ahead and record this above.
I hope you found this helpful.