# How do you differentiate g(y) =(60x^2+74)( 2x+2)  using the product rule?

Jul 14, 2017

g'(x) = color(blue)(360x^2 + 240x + 148

#### Explanation:

NOTE: I might add that the function should be of $g \left(x\right)$, not $g \left(y\right)$ (there is no $y$-variable).

We're asked to find the derivative

$\frac{\mathrm{dg}}{\mathrm{dx}} \left[\left(60 {x}^{2} + 74\right) \left(2 x + 2\right)\right]$

using the product rule.

The product rule is

$\frac{d}{\mathrm{dx}} \left[u v\right] = v \frac{\mathrm{du}}{\mathrm{dx}} + u \frac{\mathrm{dv}}{\mathrm{dx}}$

where

• $u = 2 x + 2$

• $v = 60 {x}^{2} + 74$:

$= \left(60 {x}^{2} + 74\right) \left(\frac{d}{\mathrm{dx}} \left[2 x + 2\right]\right) + \left(2 x + 2\right) \left(\frac{d}{\mathrm{dx}} \left[60 {x}^{2} + 74\right]\right)$

Applying the power rule to both terms, we have

$= \left(60 {x}^{2} + 74\right) \left(2\right) + \left(2 x + 2\right) \left(120 x\right)$

$= 120 {x}^{2} + 148 + 240 {x}^{2} + 240 x$

= color(blue)(360x^2 + 240x + 148