Q. 49 - Calculus

Question

Calculate each of the derivatives or derivative values in Exercises 47–52.

${\overline{)\frac{d^2}{d{x}^2}\left(\frac{{\displaystyle 3}}{x^{-{\displaystyle \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\sqrt{x}}\right)}}_{x=2}$

Step-by-Step Solution

Answer

${\overline{)\frac{d^2}{d{x}^2}\left(\frac{{\displaystyle 3}}{x^{-{\displaystyle \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\sqrt{x}}\right)}}_{x=2}=0$

Step 1. Given Information

We need to find the second order derivative of ${\overline{)\frac{d^2}{d{x}^2}\left(\frac{{\displaystyle 3}}{x^{-{\displaystyle \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\sqrt{x}}\right)}}_{x=2}$

Let us write $y=\left(\frac{3}{x^{-{\displaystyle \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\sqrt{x}}\right)=\frac{3}{x^{{\displaystyle \raisebox{1ex}{$-3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}+{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}=\frac{3}{x^{-1}}=3x$

Step 2. Finding First order derivative

$\frac{dy}{dx}=3$

Step 3. Finding second order derivative

$\frac{d^{2}y}{d{x}^2}=0$

at $x=2$, ${\overline{)\frac{d^2}{d{x}^2}\left(\frac{{\displaystyle 3}}{x^{-{\displaystyle \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\sqrt{x}}\right)}}_{x=2}$ is zero.