analyze the rate of change in the function: y(x) = -2x^2 + 4x - 1\n y(x) = -2x^2 - 4x - 1\n y(x) = -4x + 4\n…

Explanation:

Step1: Recall derivative formula

The derivative of a function $y = ax^{n}$ is $y'=nax^{n - 1}$, and for a sum - of - functions $y = u+v+w$, $y'=u'+v'+w'$. For $y(x)=-2x^{2}+4x - 1$, where $u=-2x^{2}$, $v = 4x$, $w=-1$.

Step2: Differentiate each term

For $u=-2x^{2}$, using the power - rule $u'=-2\times2x^{2 - 1}=-4x$. For $v = 4x$, $v'=4\times1x^{1 - 1}=4$. For $w=-1$, since the derivative of a constant is 0, $w' = 0$.

Step3: Combine the derivatives

$y'(x)=u'+v'+w'=-4x + 4+0=-4x + 4$.

Answer:

$y'(x)=-4x + 4$