Question

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

$\frac{d^2}{d{x}^2}\left({\left(5x^4-3x^2\right)}^7\left(2x^3+1\right)\right)$

Short answer

$\frac{d^2}{d{x}^2}\left({\left(5x^4-3x^2\right)}^7\left(2x^3+1\right)\right)=6x^{12}{\left(5x^2-3\right)}^5\left(7750x^7-5170x^5+3150x^4+816x^3-1925x^2+273\right)$

Step-by-step solution

Step 1. Given Information

We asks to find the second order derivative of $\frac{d^2}{d{x}^2}\left({\left(5x^4-3x^2\right)}^7\left(2x^3+1\right)\right)$

Let us write$y=\left({\left(5x^4-3x^2\right)}^7\left(2x^3+1\right)\right)$

Step 1. Finding First order derivative

$$\begin{gathered} \frac{dy}{dx}=\left(2x^3+1\right)\frac{d}{dx}{\left(5x^4-3x^2\right)}^7+{\left(5x^4-3x^2\right)}^7\frac{d}{dx}\left(2x^3+1\right) \\ \frac{dy}{dx}=7\left(2x^3+1\right){\left(5x^4-3x^2\right)}^6\frac{d}{dx}\left(5x^4-3x^2\right)+{\left(5x^4-3x^2\right)}^7\left(6x^2\right) \\ \frac{dy}{dx}=2x^{13}{\left(5x^2-3\right)}^6\left(155x^5-51x^3+70x^2-21\right) \end{gathered}$$

Step 3. Finding Second Order Derivative 

$$\begin{gathered} \frac{d^{2}y}{d{x}^2}={\left(5x^2-3\right)}^6\left(155x^5-51x^3+70x^2-21\right)\frac{d}{dx}2x^{13}+2x^{13}\left(155x^5-51x^3+70x^2-21\right)\frac{d}{dx}{\left(5x^2-3\right)}^6+2x^{13}{\left(5x^2-3\right)}^6\frac{d}{dx}\left(155x^5-51x^3+70x^2-21\right) \\ \frac{d^{2}y}{d{x}^2}=26x^{12}{\left(5x^2-3\right)}^6\left(155x^5-51x^3+70x^2-21\right)+12x^{13}\left(10x-0\right)\left(155x^5-51x^3+70x^2-21\right)+2x^{13}{\left(5x^2-3\right)}^6\left(775x^5-153x^2+140x-0\right) \\ \frac{d^{2}y}{d{x}^2}=6x^{12}{\left(5x^2-3\right)}^5\left(7750x^7-5170x^5+3150x^4+816x^3-1925x^2+273\right) \end{gathered}$$