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

Question

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

Emilia Aguilar · Accepted Answer

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

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

We are required to locate the derivative.

#(dg)/(dx) [(60x^2+74)(2x+2)]#

applying the rule of product.

The rule for products is

#d/(dx) [uv] = v(du)/(dx) + u(dv)/(dx)#

where

#u = 2x+2#

#v = 60x^2+74#:

#= (60x^2+74)(d/(dx)[2x+2]) + (2x+2)(d/(dx)[60x^2+74])#

Using both terms and the power rule, we have

#= (60x^2+74)(2) + (2x+2)(120x)#

#= 120x^2 + 148 + 240x^2 + 240x#

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

Isaiah Allen · Answer

undefined To differentiate g(y) =(60x^2+74)( 2x+2) using the product rule, follow these steps:

1. Identify the two functions being multiplied together: f(x) = 60x^2 + 74 and g(x) = 2x + 2.
2. Apply the product rule, which states that the derivative of the product of two functions is the derivative of the first function times the second function, plus the first function times the derivative of the second function.
3. Calculate the derivative of f(x) and g(x) individually.
4. Use the product rule formula to find the derivative of the given function g(x).

Here's the breakdown:

f'(x) = d/dx(60x^2 + 74) = 120x
g'(x) = d/dx(2x + 2) = 2

Now, apply the product rule:

g'(x) = f'(x) * g(x) + f(x) * g'(x)
     = (120x) * (2x + 2) + (60x^2 + 74) * 2

Therefore, the derivative of g(y) is:

g'(y) = 240x^2 + 240x + 120x^2 + 148

William Abdalla · Answer

undefined To differentiate $ g(y) = (60x^2 + 74)(2x + 2) $ using the product rule, follow these steps:

1. Identify the two functions being multiplied: $ u = 60x^2 + 74 $ and $ v = 2x + 2 $.
2. Differentiate each function with respect to $ x $ to find $ u' $ and $ v' $.
3. Apply the product rule, which states that the derivative of the product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function.
4. Use the derivatives and the product rule formula to find the derivative of the given function.

Let's denote $ u = 60x^2 + 74 $ and $ v = 2x + 2 $.

1. Find $ u' $:
$$ u' = \frac{d}{dx}(60x^2 + 74) $$
$$ = 120x $$

2. Find $ v' $:
$$ v' = \frac{d}{dx}(2x + 2) $$
$$ = 2 $$

3. Apply the product rule:
$$ g'(y) = u'v + uv' $$
$$ = (120x)(2x + 2) + (60x^2 + 74)(2) $$

4. Simplify the expression:
$$ g'(y) = 240x^2 + 240x + 120x^2 + 148 $$

$$ g'(y) = 360x^2 + 240x + 148 $$

Therefore, the derivative of $ g(y) = (60x^2 + 74)(2x + 2) $ using the product rule is $ g'(y) = 360x^2 + 240x + 148 $.