Video Transcript
Given that 𝑦 equals negative six
𝑧 squared minus 23 and 𝑧 is equal to negative four of 𝑥, determine 𝑑𝑦 by
𝑑𝑥.
We’re looking for the derivative of
𝑦 with respect to 𝑥 and we’re given 𝑦 in terms of another variable 𝑧 and 𝑧 in
terms of 𝑥. So this looks like a job for the
chain rule, which says that the derivative of 𝑦 with respect to 𝑥 is equal to the
derivative of 𝑦 with respect to some other variable 𝑧 times the derivative of 𝑧
with respect to 𝑥.
Let’s apply the chain rule to our
problem. We need to find 𝑑𝑦 by 𝑑𝑧. That is the derivative of 𝑦 with
respect to 𝑧. Luckily, we have 𝑦 written in
terms of 𝑧. So this is straightforward. 𝑦 equals negative six 𝑧 squared
minus 23. We differentiate both sides with
respect to 𝑧. 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 12𝑧.
Having found 𝑑𝑦 by 𝑑𝑧, we now
move on to finding 𝑑𝑧 by 𝑑𝑥. And to find 𝑑𝑧 by 𝑑𝑥, we use
the relation between 𝑧 and 𝑥. 𝑧 equals negative four over 𝑥 and
we can write this in exponent notation as 𝑧 equals negative four 𝑥 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, we’ll
change the 𝑧 in our rule to 𝑥. It’s important to note that we
could replace 𝑧 by any letter that we would like and it would still express the
same rule. But we choose to replace 𝑧 by 𝑥
because we’re going to be differentiating with respect to 𝑥. Applying our rule, we find 𝑑𝑧 by
𝑑𝑥 to be four 𝑥 to the negative two or if you want to avoid negative exponents
four over 𝑥 squared.
Using this expression for 𝑑𝑧 by
𝑑𝑥, we see that 𝑑𝑦 by 𝑑𝑥 is negative 12 times 𝑧 times four over 𝑥
squared. Now, it’s just a case of
simplifying this expression. And as part of the simplification,
we realize that we have 𝑑𝑦 by 𝑑𝑥 written not only in terms of 𝑥, but also in
terms of the variable 𝑧. And we’d like it to be solely in
terms of 𝑥 if that’s possible. And it is possible because 𝑧 is
negative four over 𝑥.
Substituting negative four over 𝑥
for 𝑧, we get an expression for 𝑑𝑦 by 𝑑𝑥 in terms of 𝑥 alone. And we can simplify this expression
with negative 12 times negative four times four making 192 in the numerator and 𝑥
times 𝑥 squared making 𝑥 cubed in the denominator.
𝑑𝑦 by 𝑑𝑥 is then 192 over 𝑥
cubed.
