Question

\[g(x)=x^\sqrt{x} \]
Find g'(x).

Answer 1

hint:
we can rewrite such function as below:
\[\huge g\left( x \right) = \exp \left( {\sqrt x \ln x} \right),\quad x > 0\]

Answer 2

or, using a different notation:
\[\huge g\left( x \right) = {e^{\sqrt x \ln x}},\quad x > 0\]

Answer 3

Oh okay. So afterwards we use the product rule on the exponent part?

Answer 4

yes!

Answer 5

Okay did I do this part correctly?
\[(\sqrt{x})'(lnx)+(\sqrt{x})(lnx)'=(1/2\sqrt{x})(lnx)+(\sqrt{x}/x) \]

Answer 6

that's right!
Now you have to multiply such derivative by the same function \(g(x)\)

Answer 7

Ok, thank you!