Question

The $5^{th}$ derivative of $h\left(x\right)=e^{3x+5}{x}^2$ at $x=0$ is

• A. $27$ $e^5$
• B. $54$ $e^5$
• C. $270$ $e^5$
• D. $540$ $e^5$

Answer 1

The correct option is D $540$ $e^5$

Given, $h\left(x\right)=e^{3x+5}$

We need to find $5^{th}$ derivative of $h\left(x\right)$ at $x=0$.

$h^{\prime }\left(x\right)=3e^{3x+5}{x}^2+2x{e}^{3x+5}$
$h^{\prime \prime }\left(x\right)=9e^{3x+5}{x}^2+6x{e}^{3x+5}+6x{e}^{3x+5}+2e^{3x+5}$
$h^{\prime \prime \prime }\left(x\right)=27e^{3x+5}{x}^2+18x{e}^{3x+5}+36x{e}^{3x+5}+12e^{3x+5}+6e^{3x+5}$
$h^4\left(x\right)=81e^{3x+5}{x}^2+54x{e}^{3x+5}+(54\times 3)x{e}^{3x+5}+54e^{3x+5}+54e^{3x+5}$
$h^5\left(x\right)=243e^{3x+5}{x}^2+162x{e}^{3x+5}+54e^{3x+5}+(54\times 3)e^{3x+5}+3x108e^{3x+5}$
$h^5\left(0\right)=540e^5$