Derivative of e to the Power Lnx

We know the derivative of ex, which is ex.

(ex)' = ex

We can find the derivative of elnx using chain rule.

If y = elnx, find ᵈʸ⁄d.

y = elnx

Let t = lnx.

Then, we have

y = et

Now, y = et and t = lnx. That is, y is a function of t and t is a function of x.  

By chain rule, the derivative of y with respect to x,

Substitute y = et and t = 2x.

Substitute t = lnx.

Use the identity, elnx = x.


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