### Video Transcript

Given that π¦ equals negative six
π§ squared minus 23 and π§ is equal to negative four of π₯, determine ππ¦ by
ππ₯.

Weβre looking for the derivative of
π¦ with respect to π₯ and weβre given π¦ in terms of another variable π§ and π§ in
terms of π₯. So this looks like a job for the
chain rule, which says that the derivative of π¦ with respect to π₯ is equal to the
derivative of π¦ with respect to some other variable π§ times the derivative of π§
with respect to π₯.

Letβs apply the chain rule to our
problem. We need to find ππ¦ by ππ§. That is the derivative of π¦ with
respect to π§. Luckily, we have π¦ written in
terms of π§. So this is straightforward. π¦ equals negative six π§ squared
minus 23. We differentiate both sides with
respect to π§. Using the fact that the derivative
of a difference of functions is the difference of their derivatives and using the
formula for the derivative of a number times a power with respect to the base of
that power and also the fact that the derivative of a constant function is zero, we
get negative 12π§.

Having found ππ¦ by ππ§, we now
move on to finding ππ§ by ππ₯. And to find ππ§ by ππ₯, we use
the relation between π§ and π₯. π§ equals negative four over π₯ and
we can write this in exponent notation as π§ equals negative four π₯ to the negative
one. We can now apply our rule about the
derivative of a number times a power of a variable with respect to that
variable.

To make things clearer, weβll
change the π§ in our rule to π₯. Itβs important to note that we
could replace π§ by any letter that we would like and it would still express the
same rule. But we choose to replace π§ by π₯
because weβre going to be differentiating with respect to π₯. Applying our rule, we find ππ§ by
ππ₯ to be four π₯ to the negative two or if you want to avoid negative exponents
four over π₯ squared.

Using this expression for ππ§ by
ππ₯, we see that ππ¦ by ππ₯ is negative 12 times π§ times four over π₯
squared. Now, itβs just a case of
simplifying this expression. And as part of the simplification,
we realize that we have ππ¦ by ππ₯ written not only in terms of π₯, but also in
terms of the variable π§. And weβd like it to be solely in
terms of π₯ if thatβs possible. And it is possible because π§ is
negative four over π₯.

Substituting negative four over π₯
for π§, we get an expression for ππ¦ by ππ₯ in terms of π₯ alone. And we can simplify this expression
with negative 12 times negative four times four making 192 in the numerator and π₯
times π₯ squared making π₯ cubed in the denominator.

ππ¦ by ππ₯ is then 192 over π₯
cubed.