Question

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

$\frac{d^2}{d{x}^2}\left({\left(x\sqrt{x+1}\right)}^{-2}\right)$

Short answer

The second order derivatives of$\frac{d^2}{d{x}^2}\left({\left(x\sqrt{x+1}\right)}^{-2}\right)$islocalid="1649329655183" $\frac{2\left(6x^2+8x+3\right)}{x^4{\left(x+1\right)}^3}$

Step-by-step solution

Step 1. Given Information

We need to find $\frac{d^2}{d{x}^2}\left({\left(x\sqrt{x+1}\right)}^{-2}\right)$

Let us write

$$\begin{gathered} y=\left({\left(x\sqrt{x+1}\right)}^{-2}\right) \\ y=\frac{1}{x^2{\left(\sqrt{x+1}\right)}^2}=\frac{1}{x^2(x+1)} \end{gathered}$$

Step 2. Finding First Derivative

Differentiating with respect to $x$

localid="1649329091115" role="math" $$\begin{gathered} \frac{dy}{dx}=\frac{0x^2\left(x+1\right)-\left[2x\left(x+1\right)-x^2\right]}{x^4{\left(x+1\right)}^2} \\ \frac{dy}{dx}=\frac{-2x^2-2x+x^2}{x^4{\left(x+1\right)}^2}=\frac{-x^2-2x}{x^4{\left(x+1\right)}^2} \end{gathered}$$

Step 3. Finding Second order Derivative

$$\begin{gathered} \frac{d^{2}y}{d{x}^2}=\frac{(-2x-2)\left(x^4\left(x+1\right)\right)-\left(-x^2-2x\right)\left(4x^3{\left(x+1\right)}^2+x^4\times 2\left(x+1\right)\right)}{x^8{\left(x+1\right)}^2} \\ \frac{d^{2}y}{d{x}^2}=\frac{(-2x-2)\left(x^5+x^4\right)+\left(x^2+2x\right)\left(4x^5+8x^4+4x^3\right)+2x^5+2x^4}{x^8{\left(x+1\right)}^2} \\ \frac{d^{2}y}{d{x}^2}=\frac{2\left(6x^2+8x+3\right)}{x^4{\left(x+1\right)}^3} \end{gathered}$$