# Question Video: Deriving Exponential Functions Mathematics • Higher Education

Given that 𝑥 = 𝑒^𝑦, find d𝑦/d𝑥, giving your answer in terms of 𝑥.

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### Video Transcript

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.