How do you differentiate #g(y) =(60x^2+74)( 2x+2) # using the product rule?
We are required to locate the derivative.
applying the rule of product.
The rule for products is
where
Using both terms and the power rule, we have
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To differentiate g(y) =(60x^2+74)( 2x+2) using the product rule, follow these steps:
 Identify the two functions being multiplied together: f(x) = 60x^2 + 74 and g(x) = 2x + 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.
 Calculate the derivative of f(x) and g(x) individually.
 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
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To differentiate ( g(y) = (60x^2 + 74)(2x + 2) ) using the product rule, follow these steps:
 Identify the two functions being multiplied: ( u = 60x^2 + 74 ) and ( v = 2x + 2 ).
 Differentiate each function with respect to ( x ) to find ( u' ) and ( v' ).
 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.
 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 ).

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

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

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

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 ).
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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