What is the derivative of #x^(lnx)#?
The derivative of
There are no rules that we can apply to easily differentiate this equation, so we just have to mess with it until we find an answer.
If we take the natural log of both sides, we are changing the equation. We can do this as long as we take into account that this will be a completely new equation:
Differentiate both sides:
Okay, now we're done messing with that equation. Let's go back to the original problem:
We can rewrite this as
Now, let's differentiate this using the exponent rule:
Conveniently, we already found the first term above, so we can easily simplify this.
Take Natural logs of both sides
Differentiate (implicitly) wrt