Question Video: Deriving Exponential Functions Mathematics • Higher Education

Given that ๐‘ฅ = ๐‘’^๐‘ฆ, find d๐‘ฆ/d๐‘ฅ, giving your answer in terms of ๐‘ฅ.

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Video Transcript

Given that ๐‘ฅ is equal to ๐‘’ to the power of ๐‘ฆ, find d๐‘ฆ by d๐‘ฅ, giving your answer in terms of ๐‘ฅ.

We can start by differentiating ๐‘ฅ with respect to ๐‘ฆ. Using the fact that the differential of an exponential is just the exponential, we obtain that d๐‘ฅ by d๐‘ฆ is equal to ๐‘’ to the power of ๐‘ฆ. Now, weโ€™re trying to find the differential of the reciprocal function, so thatโ€™s ๐‘ฆ, with respect to ๐‘ฅ. So thatโ€™s d๐‘ฆ by d๐‘ฅ. 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 d๐‘ฆ by d๐‘ฅ is equal to one over d๐‘ฅ by d๐‘ฆ. In order to use this, we must ensure that the denominator of our fraction is nonzero. So thatโ€™s d๐‘ฅ by d๐‘ฆ.

Weโ€™ve just found that d๐‘ฅ by d๐‘ฆ is equal to ๐‘’ to the power of ๐‘ฆ. Since ๐‘’ to the ๐‘ฆ is an exponential, we know that ๐‘’ to the power of ๐‘ฆ is going to be greater than zero for all values of ๐‘ฆ. Therefore, it is nonzero. And so, weโ€™re able to use this formula. And so, we obtain that d๐‘ฆ by d๐‘ฅ is equal to one over ๐‘’ to the power of ๐‘ฆ. However, the question has asked us to give our answer in terms of ๐‘ฅ. In order to get our answer in terms of ๐‘ฅ, we can use the fact that ๐‘ฅ is equal to ๐‘’ to the power of ๐‘ฆ and substitute ๐‘ฅ in for ๐‘’ to the power of ๐‘ฆ. From here, we reach our solution, which is that d๐‘ฆ by d๐‘ฅ is equal to one over ๐‘ฅ.

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