1. ## derivativ

find f prim (1) when f(x)= x^1+sqrt x .

thanks

2. Hi

$\displaystyle f(x) = x^{1+\sqrt{x}} = e^{(1+\sqrt{x})ln(x)}$

The first derivative of

$\displaystyle e^{u(x)}$ is $\displaystyle u'(x) e^{u(x)}$

Here $\displaystyle u(x) = (1+\sqrt{x})ln(x)$

u(x) is a product of 2 functions
$\displaystyle u'(x) = \frac{1}{2 \sqrt{x}}ln(x) + \frac{1+\sqrt{x}}{x}$

3. hi !
I dont know this is power fuction or exponetial function can you explain more .
thank you

4. familiar with logarithmic differentiation?

$\displaystyle y = x^{1+\sqrt{x}}$

$\displaystyle \ln{y} = \ln\left(x^{1+\sqrt{x}}\right)$

$\displaystyle \ln{y} = (1 + \sqrt{x})\ln{x}$

$\displaystyle \frac{d}{dx}[\ln{y} = (1 + \sqrt{x})\ln{x}]$

$\displaystyle \frac{y'}{y} = (1 + \sqrt{x})\cdot \frac{1}{x} + \ln{x} \cdot \frac{1}{2\sqrt{x}}$

$\displaystyle y' = y\left((1 + \sqrt{x})\cdot \frac{1}{x} + \ln{x} \cdot \frac{1}{2\sqrt{x}}\right)$

$\displaystyle y' = x^{1+\sqrt{x}}\left((1 + \sqrt{x})\cdot \frac{1}{x} + \ln{x} \cdot \frac{1}{2\sqrt{x}}\right)$