### Video Transcript

Given that π¦ equals negative seven [π‘]
cubed plus eight and π§ equals negative seven [π‘] squared plus three, find the rate of
change of π¦ with respect to π§.

When faced with a question about
the rate of change of something, we should be thinking about derivatives. Here we want to find the rate of
change of π¦ with respect to π§. So weβre going to work out dπ¦ by
dπ§. Thatβs the first derivative of π¦
with respect to π§.

We then recall that, given two
parametric equations β π₯ is equal to π of π‘ and π¦ is equal to π of π‘ β we find
dπ¦ by dπ₯ by multiplying dπ¦ by dπ‘ by one over dπ₯ by dπ‘. Or equivalently, by dividing dπ¦ by
dπ‘ by dπ₯ by dπ‘. In this example, our two functions
are π¦ and π§. So we say that dπ¦ by dπ§ equals
dπ¦ by dπ‘ divided by dπ§ by dπ‘. And we see that weβre going to need
to begin by differentiating each function with respect to π‘.

Weβll begin by differentiating π¦
with respect to π‘. Remember, to differentiate a
polynomial term, we multiply the term by the exponent and then reduce that exponent
by one. So the first derivative of negative
70 cubed is three times negative 70 squared. And actually, the first derivative
of eight is zero. Of course, we donβt really need to
include that plus zero. So we find that dπ¦ by dπ‘ is equal
to negative 21π‘ squared.

Weβll now repeat this for dπ§ by
dπ‘. This time, the first derivative is
two times negative seven π‘, which is negative 14π‘. dπ¦ by dπ§ is what we get when
we divide dπ¦ by dπ‘ by dπ§ by dπ‘. So thatβs negative 21π‘ squared
divided by negative 14π‘. Of course, a negative divided by a
negative is a positive. And we can divide both the
numerator and the denominator by π‘. Our final step is to simplify by
dividing by 21 and 14 by seven. So we find the rate of change of π¦
with respect to π§ to be three π‘ over two.