How do you differentiate #g(y) =(2x-5 )(x^3 + 4) # using the product rule?
Applying the rule of product
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To differentiate ( g(y) = (2x - 5)(x^3 + 4) ) using the product rule, follow these steps:
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Identify the two functions: ( u = 2x - 5 ) and ( v = x^3 + 4 ).
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Compute the derivatives of ( u ) and ( v ): ( u' = 2 ) and ( v' = 3x^2 ).
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Apply the product rule: ( g'(x) = u'v + uv' ).
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Substitute the derivatives and the original functions into the product rule formula:
[ g'(x) = (2)(x^3 + 4) + (2x - 5)(3x^2) ]
Simplify the expression.
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( g'(x) = 2x^3 + 8 + 6x^3 - 15x^2 )
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Combine like terms to get the final derivative:
[ g'(x) = 8x^3 - 15x^2 + 8 ]
That's the derivative of ( g(y) ) using the product rule.
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When evaluating a one-sided 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 one-sided 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 one-sided 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 one-sided 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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