Video Transcript
Given that π₯ is equal to π to the
power of π¦, find dπ¦ by dπ₯, giving your answer in terms of π₯.
We can start by differentiating π₯
with respect to π¦. Using the fact that the
differential of an exponential is just the exponential, we obtain that dπ₯ by dπ¦ is
equal to π to the power of π¦. Now, weβre trying to find the
differential of the reciprocal function, so thatβs π¦, with respect to π₯. So thatβs dπ¦ by dπ₯. And in order to do this, we can use
the fact that the derivative of an inverse of a function is equal to the reciprocal
of the derivative of the function. Giving us that dπ¦ by dπ₯ is equal
to one over dπ₯ by dπ¦. In order to use this, we must
ensure that the denominator of our fraction is nonzero. So thatβs dπ₯ by dπ¦.
Weβve just found that dπ₯ by dπ¦ is
equal to π to the power of π¦. Since π to the π¦ is an
exponential, we know that π to the power of π¦ is going to be greater than zero for
all values of π¦. Therefore, it is nonzero. And so, weβre able to use this
formula. And so, we obtain that dπ¦ by dπ₯
is equal to one over π to the power of π¦. However, the question has asked us
to give our answer in terms of π₯. In order to get our answer in terms
of π₯, we can use the fact that π₯ is equal to π to the power of π¦ and substitute
π₯ in for π to the power of π¦. From here, we reach our solution,
which is that dπ¦ by dπ₯ is equal to one over π₯.
