How do you differentiate #f(x)=(4e^x+1)(x3)# using the product rule?
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To differentiate ( f(x) = (4e^x + 1)(x  3) ) using the product rule, follow these steps:

Identify the functions ( u(x) ) and ( v(x) ). Let ( u(x) = 4e^x + 1 ) and ( v(x) = x  3 ).

Compute the derivatives ( u'(x) ) and ( v'(x) ). ( u'(x) = 4e^x ) and ( v'(x) = 1 ).

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

Substitute the derivatives and the functions: ( f'(x) = (4e^x)(x  3) + (4e^x + 1)(1) ).

Simplify the expression: ( f'(x) = 4xe^x  12e^x + 4e^x + 1 ).

Combine like terms: ( f'(x) = 4xe^x  8e^x + 1 ).
So, the derivative of ( f(x) = (4e^x + 1)(x  3) ) with respect to ( x ) using the product rule is ( f'(x) = 4xe^x  8e^x + 1 ).
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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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