Question

find the fully simplified derivative of $y = 4x^{2}+3x + 6x^{-2}$
$y = 8x + 3 + 12/x$
$y = 8x + 3 + 12/x^{3}$
$y = 8x + 3 - 12/x^{3}$
$y = 8x + 3 - 12x^{-3}$

Explanation:

Step1: Apply the power rule

The power rule states that if \(y = ax^n\), then \(y'=nax^{n - 1}\).
For \(y = 4x^2\), using the power rule: \(y'_1=2\times4x^{2 - 1}=8x\).
For \(y = 3x\), using the power rule: \(y'_2 = 3\times1x^{1 - 1}=3\).
For \(y = 6x^{- 2}\), using the power rule: \(y'_3=-2\times6x^{-2 - 1}=-12x^{-3}=-\frac{12}{x^{3}}\).

Step2: Combine the derivatives

\(y'=y'_1 + y'_2 + y'_3\).
So \(y'=8x + 3-\frac{12}{x^{3}}\).

Answer:

\(y' = 8x + 3-\frac{12}{x^{3}}\) (the third option).