Question

Finding Higher-Order Derivatives In Exercise, find the second derivative of the function.

Answer 1

Answer: The second derivative of the function is $f''(x)=\dfrac{1}{2x}$

Step-by-step explanation:

Since we have given that

$f(x)=x\ln \sqrt{x}+2x$

We need to find the second derivative of the function.

So, the first derivative would be

$f'(x)=1\times \ln\sqrt{x}+x\dfrac{1}{\sqrt{x}}\times \dfrac{1}{2\sqrt{x}}+2\\\\f'(x)=\ln \sqrt{x}+\dfrac{1}{2}+2\\\\f'(x)=\ln\sqrt{x}+\dfrac{3}{2}$

Now, second derivative would be

$f''(x)=\dfrac{1}{\sqrt{x}}\times \dfrac{1}{2\sqrt{x}}\\\\f''(x)=\dfrac{1}{2x}$

Hence, the second derivative of the function is $f''(x)=\dfrac{1}{2x}$