Question

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

Answer 1

`1`

Explanation:

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`