Answer: The second derivative of the function is [tex]f''(x)=\dfrac{1}{2x}[/tex]
Step-by-step explanation:
Since we have given that
[tex]f(x)=x\ln \sqrt{x}+2x[/tex]
We need to find the second derivative of the function.
So, the first derivative would be
[tex]f'(x)=1\times \ln\sqrt{x}+x\dfrac{1}{\sqrt{x}}\times \dfrac{1}{2\sqrt{x}}+2\\\\f'(x)=\ln \sqrt{x}+\dfrac{1}{2}+2\\\\f'(x)=\ln\sqrt{x}+\dfrac{3}{2}[/tex]
Now, second derivative would be
[tex]f''(x)=\dfrac{1}{\sqrt{x}}\times \dfrac{1}{2\sqrt{x}}\\\\f''(x)=\dfrac{1}{2x}[/tex]
Hence, the second derivative of the function is [tex]f''(x)=\dfrac{1}{2x}[/tex]
