In details

Derivatives of Exponential Functions


To get a formula for the derivative of exponential functions

y =

we rewrite this equation as

x =

and differentiate implicitly using to get

that we can rewrite using y = as

Thus, showing that if is a differentiable function, so its derivative with respect to x é

In the special case where b = e we have 1n and = 1n, like this becomes

In addition, if u is a differentiable function of xthen one has from and what

NOTE.It is important to distinguish between (variable exponent and constant basis) and (variable base and constant exponent).

Example

The following calculations use

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