How do you differentiate #y = (x + 7)^10 (x^2 + 2)^7#?
See the explanation section, below.
Consequently,
Now, simplify as desired using algebra.
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To differentiate ( y = (x + 7)^{10} (x^2 + 2)^7 ), you can use the product rule and chain rule of differentiation. Here are the steps:

Apply the product rule: ( (uv)' = u'v + uv' ), where ( u = (x + 7)^{10} ) and ( v = (x^2 + 2)^7 ).

Differentiate ( u = (x + 7)^{10} ) with respect to ( x ) using the chain rule: ( u' = 10(x + 7)^9 ).

Differentiate ( v = (x^2 + 2)^7 ) with respect to ( x ) using the chain rule: ( v' = 14(x^2 + 2)^6 (2x) ).

Substitute the derivatives and original functions back into the product rule formula: [ y' = (10(x + 7)^9)(x^2 + 2)^7 + (x + 7)^{10}[14(x^2 + 2)^6 (2x)] ]

Simplify the expression if needed, and that will be the derivative of the given function.
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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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