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

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

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

Answer

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).