Functions differentiable and its derivative

Question

Where is the function $f(x) = \sqrt[3]{x}e^{\sqrt{x}}$ defined, where is it differentiable, and what is its derivative

Could anyone help me with this? I am not good at this part of math theory?

Answer 1

The author of the function must specify the domain. If not, you can assume that is defined for $x>0$, being $x$ a real number. On the other hand, basic calculus shows that this function is differentiable in its domain. Finally, the derivative can be computed with the usual derivatives rules and you can check (using wolframalpha, for example) that the result is $$ =\frac{e^{\sqrt{x}}(3 \sqrt{x}+2)}{6 x^{2 / 3}} $$