How do you differentiate #f(x)=x^2(x+7)^3# using the product rule?
In this instance, the chain rule must be applied in order to distinguish the terms "A" and "B".
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# :. f'(x) = x(x+7)^2 (5x+14} #
The Product Rule for Differentiation is something you should learn and practice using if you are studying math:
"The first times the derivative of the second plus the second times the derivative of the first" is the rule that I was taught to memorize.
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To differentiate ( f(x) = x^2(x+7)^3 ) using the product rule, you can follow these steps:

Identify the functions ( u ) and ( v ) where ( f(x) = u(x) \cdot v(x) ). Let ( u(x) = x^2 ) and ( v(x) = (x+7)^3 ).

Apply the product rule, which states that the derivative of the product of two functions ( u ) and ( v ) is given by: [ \frac{d}{dx}[u(x) \cdot v(x)] = u'(x) \cdot v(x) + u(x) \cdot v'(x) ]

Differentiate each function separately: [ u'(x) = 2x ] [ v'(x) = 3(x+7)^2 ]

Substitute the derivatives and the original functions into the product rule formula: [ f'(x) = u'(x) \cdot v(x) + u(x) \cdot v'(x) ] [ f'(x) = (2x) \cdot (x+7)^3 + (x^2) \cdot 3(x+7)^2 ]

Simplify the expression: [ f'(x) = 2x(x+7)^3 + 3x^2(x+7)^2 ]
This is the derivative of the function ( f(x) = x^2(x+7)^3 ) using the product rule.
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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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