How do you differentiate #y=(5x^5+5)(2x^53)# using the product rule?
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To differentiate y=(5x^5+5)(2x^53) using the product rule, follow these steps:

Identify the two functions being multiplied: ( u = 5x^5 + 5 ) and ( v = 2x^5  3 ).

Apply the product rule, which states that the derivative of the product of two functions ( u ) and ( v ) is given by ( u'v + uv' ).

Find the derivatives of ( u ) and ( v ): ( u' = 25x^4 ) and ( v' = 10x^4 ).

Substitute these values into the product rule formula: ( y' = (25x^4)(2x^5  3) + (5x^5 + 5)(10x^4) ).

Simplify the expression: ( y' = 50x^9  75x^4 + (50x^9  50x^4) ).

Combine like terms: ( y' = 100x^9  125x^4 ).
Therefore, the derivative of ( y = (5x^5 + 5)(2x^5  3) ) using the product rule is ( y' = 100x^9  125x^4 ).
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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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