Home > Calculus > Differentiating Exponential Functions with Base e

What is the derivative of #e^(lnx)#?

Answer 1

Luke Alvarez

#1#

We can also do this without first using the identity #e^lnx=x#, although we will have to use this eventually.

Note that #d/dxe^x=e^x#, so when we have a function in the exponent the chain rule will apply: #d/dxe^u=e^u*(du)/dx#.

So:

#d/dxe^lnx=e^lnx(d/dxlnx)#

The derivative of #lnx# is #1/x#:

#d/dxe^lnx=e^lnx(1/x)#

Then using the identity #e^lnx=x#:

#d/dxe^lnx=x(1/x)=1#

Which is the same as the answer we'd get if we use the identity from the outset (which is what I recommend you do--this is just a fun way to show that "calculus works".)

#d/dxe^lnx=d/dxx=1#

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Answer 2

Adam Adkins

The derivative of ( e^{\ln(x)} ) is ( \frac{1}{x} ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.