5. differentiate e^(9x^2 + 2x + 1). (18x+2)e^(9x^2+2x+1) output: (18x + 2)e^(9x^2 + 2x + 1) correct! 6…

Explanation:

Step1: Apply product - rule

The product - rule states that if $y = uv$, where $u$ and $v$ are functions of $x$, then $y^\prime=u^\prime v + uv^\prime$. Let $u = 9x^{2}+x$ and $v = e^{x^{2}+7x}$. First, find $u^\prime$ and $v^\prime$. $u^\prime=\frac{d}{dx}(9x^{2}+x)=18x + 1$ $v^\prime=\frac{d}{dx}(e^{x^{2}+7x})=e^{x^{2}+7x}(2x + 7)$ (by the chain - rule, since if $y = e^{f(x)}$, then $y^\prime=e^{f(x)}f^\prime(x)$)

Step2: Calculate the derivative

Using the product - rule $y^\prime=u^\prime v+uv^\prime$, we have: $y^\prime=(18x + 1)e^{x^{2}+7x}+(9x^{2}+x)e^{x^{2}+7x}(2x + 7)$ $y^\prime=e^{x^{2}+7x}[(18x + 1)+(9x^{2}+x)(2x + 7)]$ $y^\prime=e^{x^{2}+7x}(18x + 1+18x^{3}+63x^{2}+2x^{2}+7x)$ $y^\prime=e^{x^{2}+7x}(18x^{3}+65x^{2}+25x + 1)$

Answer:

$e^{x^{2}+7x}(18x^{3}+65x^{2}+25x + 1)$