Question

1. find $\frac{d^{2}y}{dx^{2}}$ for $y = \frac{2}{x^{2}}+e^{x}+(x^{5}+x + 1)(x + 3)$.

Explanation:

Step1: Rewrite the function

Rewrite $y=\frac{2}{x^{2}}+e^{x}+(x^{5}+x + 1)(x + 3)$ as $y = 2x^{-2}+e^{x}+x^{6}+3x^{5}+x^{2}+3x+x + 3=2x^{-2}+e^{x}+x^{6}+3x^{5}+x^{2}+4x + 3$.

Step2: Find the first - derivative

Using the power rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$ and $\frac{d}{dx}(e^{x})=e^{x}$, we have $y'=-4x^{-3}+e^{x}+6x^{5}+15x^{4}+2x + 4$.

Step3: Find the second - derivative

Again applying the power rule and $\frac{d}{dx}(e^{x})=e^{x}$, we get $y'' = 12x^{-4}+e^{x}+30x^{4}+60x^{3}+2$.

Answer:

$y''=\frac{12}{x^{4}}+e^{x}+30x^{4}+60x^{3}+2$