### Video Transcript

Determine the rate of change of the
function π of π₯ is equal to five divided by nine π₯ at π₯ is equal to the square
root of two.

The question wants us to find the
rate of change of the function π of π₯ when π₯ is equal to the square root of
two. The rate of change of our function
π of π₯ when π₯ is equal to π is given by π prime of π. And we say this is equal to the
limit as β approaches zero of π evaluated at π plus β minus π evaluated at π
divided by β if this limit exists. Since we want to find the rate of
change when π₯ is equal to the square root of two, weβll set π equal to the square
root of two and π of π₯ to be five divided by nine π₯.

We can do this in steps. Letβs start by simplifying the
numerator inside of our limit. With π equal to the square root of
two, we get π evaluated at the square root of two plus β minus π evaluated at the
square root of two. So we want to evaluate π at the
square root of two plus β and then subtract π evaluated at the square root of
two. And, remember, π of π₯ is five
divided by nine π₯. So π of π₯ evaluated at the square
root of two plus β is equal to five divided by nine times the square root of two
plus β. Similarly, π evaluated at the
square root of two is five divided by nine root two.

We now want to simplify this
expression. We can do this by using cross
multiplication. When we cross multiply these two
fractions, in our denominator, weβll get the product of the two denominators. Thatβs nine times root two plus β
multiplied by nine root two. And we get the first term in our
numerator by multiplying our first numerator by our second denominator. Thatβs five multiplied by nine root
two. Then we want to subtract the second
term, which will be the numerator of our second fraction multiplied by the
denominator in our first fraction. Thatβs five times nine times the
square root of two plus β.

And we can simplify the numerator
in this expression. Five times nine is equal to 45. Multiplying out and simplifying our
second term, we get 45 root two plus 45β. So this gives us a numerator of 45
root two minus 45 root two plus 45β. And we can simplify this further,
since 45 root two minus 45 root two is equal to zero. This means our numerator is just
negative 45β. And we can simplify this even
further. 45 divided by nine is equal to
five.

So weβve found the following
expression for π evaluated at the square root of two plus β minus π evaluated at
the square root of two. So we can substitute this into our
limit for π prime of root two. Thatβs the limit as β approaches
zero of π evaluated at root two plus β minus π evaluated at root two divided by
β. Substituting in our expression for
the numerator, we get the limit as β approaches zero of negative five β divided by
the square root of two plus β times nine times the square root of two all divided by
β.

Now, weβre ready to simplify this
expression. First, instead of dividing through
by β, weβre going to multiply through by one divided by β. Next, weβll cancel the shared
factor of β in our numerator and our denominator. This gives us the limit as β
approaches zero of negative five divided by the square root of two plus β times nine
times root two. And this is just the limit of a
rational function. We can attempt to do this by direct
substitution.

Substituting β is equal to zero, we
get negative five divided by root two plus zero multiplied by nine root two. And we can then just calculate this
expression to get negative five divided by 18. Therefore, weβve shown from first
principles the rate of change of the function π of π₯ is equal to five divided by
nine π₯ where π₯ is equal to root two is given by negative five divided by 18.