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How do you differentiate #f(x)=(4e^x+1)(x-3)# using the product rule?

Answer 1

Christopher Agresta

#4xe^x -8e^x +1#

It would be #(x-3)d/dx(4e^x +1) + (4e^x +1)d/dx (x-3)#

= #4e^x (x-3)+ 4e^x +1 #

= #4xe^x -8e^x +1#

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Answer 2

Paisley Johnson

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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Answer from HIX Tutor

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