Question

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

${\overline{)\frac{d^3}{d{x}^3}\left(x{\left(3x^2+1\right)}^9\right)}}_{x=0}$

Short answer

${\overline{)\frac{d^3}{d{x}^3}\left(x{\left(3x^2+1\right)}^9\right)}}_{x=0}=162$

Step-by-step solution

Step 1. Given Information

We need to find${\overline{)\frac{d^3}{d{x}^3}\left(x{\left(3x^2+1\right)}^9\right)}}_{x=0}$

Let us write $y=x{\left(3x^2+1\right)}^9$

Step 2. Finding First derivative

$$\begin{gathered} \frac{dy}{dx}={\left(3x^2+1\right)}^9\frac{d}{dx}x+x\frac{d}{dx}{\left(3x^2+1\right)}^9 \\ \frac{dy}{dx}={\left(3x^2+1\right)}^9+9x{\left(3x^2+1\right)}^8\left(6x\right) \\ \frac{dy}{dx}={\left(3x^2+1\right)}^9+54x^2{\left(3x^2+1\right)}^8 \\ \frac{dy}{dx}={\left(3x^2+1\right)}^8\left(57x^2+1\right) \end{gathered}$$

Step 2. Finding Second Order Derivative

$$\begin{gathered} \frac{d^{2}y}{d{x}^2}=8{\left(3x^2+1\right)}^7\left(6x\right)+{\left(3x^2+1\right)}^8\left(114x\right) \\ \frac{d^{2}y}{d{x}^2}=48x\left({\left(3x^2+1\right)}^7\right)+114x{\left(3x^2+1\right)}^8 \\ \frac{d^{2}y}{d{x}^2}=162x{\left(3x^2+1\right)}^7\left(19x^2+1\right) \end{gathered}$$

Step 4. Finding third order derivative

$$\begin{gathered} \frac{d^{3}y}{d{x}^3}=162{\left(3x^2+1\right)}^7\left(19x^2+1\right)+162x\times 7{\left(3x^2+1\right)}^6\left(19x^2+1\right)+162x{\left(3x^2+1\right)}^7\left(38x\right) \\ \frac{d^{3}y}{d{x}^3}=162{\left(3x^2+1\right)}^6\left(969x^4+102x^2+1\right) \end{gathered}$$

Step 5. Finding third derivative at x=0

$\frac{d^{3}y}{d{x}^3}=162{\left(0^2+1\right)}^6\left(969\times 0^4+102\times 0^2+1\right)=162\times 1=162$