How do you differentiate # f(x)=(2x+1)(4x^2)(1+x^2) # using the product rule?
You could further simply it if you are required to do so.
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To differentiate ( f(x)=(2x+1)(4x^2)(1+x^2) ) using the product rule, follow these steps:

Identify the functions ( u ) and ( v ): ( u = 2x + 1 ) ( v = (4  x^2)(1 + x^2) )

Apply the product rule: ( f'(x) = u'v + uv' )

Differentiate ( u ) and ( v ): ( u' = 2 ) ( v' = (4  x^2)'(1 + x^2) + (4  x^2)(1 + x^2)' )

Simplify ( u' ) and ( v' ): ( (4  x^2)' = 2x ) ( (1 + x^2)' = 2x )

Substitute ( u' ) and ( v' ) into the product rule: ( f'(x) = 2(4  x^2)(1 + x^2) + (2x + 1)(2x) + (4  x^2)(1 + x^2)(2x) )

Expand and simplify the expression: ( f'(x) = 2(4  x^2)(1 + x^2)  2x(2x + 1) + 2x(4  x^2)(1 + x^2) )

Combine like terms if necessary.
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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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