### Video Transcript

Find the derivative of the function
π of π§ equals negative three π to the four π§ over four π§ plus one.

Here we have a function of a
function or a composite function. This tells us weβre going to need
to apply the chain rule. This says that the derivative of π
of π of π₯ is equal to the derivative of π of π of π₯ multiplied by the
derivative of π of π₯. Alternatively, we can say that if π¦
is equal to this composite function, π of π of π₯, then if we let π’ be equal to
π of π₯, then π¦ is equal to π of π’. And this means we can say that the
derivative of π¦ with respect to π₯ is equal to the derivative of π¦ with respect to
π’ multiplied by the derivative of π’ with respect to π₯.

In this example, we say that π¦ is
equal to negative three times π to the power of π’, where π’ is equal to four π§
over four π§ plus one. And since weβre going to be
differentiating with respect to π§, we alter the formula slightly. And we say that dπ¦ by dπ§ is equal
to dπ¦ by dπ’ times dπ’ by dπ§. So weβre going to need to work out
dπ¦ by dπ’ and dπ’ by dπ§. dπ¦ by dπ’ is a fairly easy one to differentiate. We know that the derivative of π
to the power of π’ is π to the power of π’. So the derivative of negative three
π to the power of π’ is negative three π to the power of π’. And then, we can replace π’ with
four π§ over four π§ plus one to get negative three π to the four π§ over four π§
plus one. But what about the derivative of four
π§ over four π§ plus one?

Well, here, we need to use the
quotient rule. This says that the derivative of π
of π₯ over π of π₯ is equal to the function π of π₯ times the derivative of π of
π₯ minus the function π of π₯ times the derivative of π of π₯. And thatβs all over π of π₯
squared. We change π of π₯ to π of π§ and
π of π₯ to π of π§. Then, the derivative of the
numerator of our fraction is four. And the derivative of the
denominator of our fraction is also four. So the equivalent to π of π§ times
the derivative of π of π§ is four π§ plus one times four. And the equivalent to π of π§
times the derivative of π of π§ is four π§ times four. And thatβs all over the denominator
squared. Thatβs four π§ plus one
squared. Now, distributing the parentheses
and we simply end up with four on the numerator of this fraction. So dπ’ by dπ§ is equal to four over
four π§ plus one squared.

Letβs now substitute everything we
have into the formula for the chain rule. dπ¦ by dπ’ times dπ’ by dπ§ is negative
three π to the four π§ over four π§ plus one times four over four π§ plus one
squared. And if we simplify, we see that the
derivative of our function is negative 12π to the four π§ over four π§ plus one all
over four π§ plus one squared.