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

Answer 1

William Abdalla

#(d f(x))/(d x)=3*e^x* x^2+e^x*(x^3-1) *l n (e)#

#f(x)=e^x(x^3-1)#

#(d f(x))/(d x)=(e^x)'*(x^3-1)+(x^3-1)^'*e^x#

#(d f(x))/(d x)=e^x*l n e*(x^3-1)+3x^2*e^x#

#(d f(x))/(d x)=3*e^x* x^2+e^x*(x^3-1) *l n (e)#

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Answer 2

Emma Aldridge

To differentiate ( f(x) = e^x(x^3 - 1) ) using the product rule, follow these steps:

Identify the two functions being multiplied: ( u(x) = e^x ) and ( v(x) = x^3 - 1 ).
Use the product rule formula: ( f'(x) = u'(x)v(x) + u(x)v'(x) ).
Find the derivatives of ( u(x) ) and ( v(x) ):
- ( u'(x) = e^x ) (derivative of ( e^x ) is itself).
- ( v'(x) = 3x^2 ) (derivative of ( x^3 - 1 ) is ( 3x^2 )).
Substitute the derivatives and original functions into the product rule formula:
- ( f'(x) = e^x(x^3 - 1) + e^x(3x^2) ).
Simplify the expression if necessary:
- ( f'(x) = e^x(x^3 - 1 + 3x^2) ).
- ( f'(x) = e^x(x^3 + 3x^2 - 1) ).

So, the derivative of ( f(x) = e^x(x^3 - 1) ) using the product rule is ( f'(x) = e^x(x^3 + 3x^2 - 1) ).

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Answer from HIX Tutor

When evaluating a one-sided 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.