Given that 𝑥 is equal to 𝑒 to the
power of 𝑦, find d𝑦 by d𝑥, giving your answer in terms of 𝑥.
We can start by differentiating 𝑥
with respect to 𝑦. Using the fact that the
differential of an exponential is just the exponential, we obtain that d𝑥 by d𝑦 is
equal to 𝑒 to the power of 𝑦. Now, we’re trying to find the
differential of the reciprocal function, so that’s 𝑦, with respect to 𝑥. So that’s d𝑦 by d𝑥. 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 d𝑦 by d𝑥 is equal
to one over d𝑥 by d𝑦. In order to use this, we must
ensure that the denominator of our fraction is nonzero. So that’s d𝑥 by d𝑦.
We’ve just found that d𝑥 by d𝑦 is
equal to 𝑒 to the power of 𝑦. Since 𝑒 to the 𝑦 is an
exponential, we know that 𝑒 to the power of 𝑦 is going to be greater than zero for
all values of 𝑦. Therefore, it is nonzero. And so, we’re able to use this
formula. And so, we obtain that d𝑦 by d𝑥
is equal to one over 𝑒 to the power of 𝑦. However, the question has asked us
to give our answer in terms of 𝑥. In order to get our answer in terms
of 𝑥, we can use the fact that 𝑥 is equal to 𝑒 to the power of 𝑦 and substitute
𝑥 in for 𝑒 to the power of 𝑦. From here, we reach our solution,
which is that d𝑦 by d𝑥 is equal to one over 𝑥.