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

Answer 1

Isaiah Allen

#f'(x) = e^x (x^3 - 1) + e^x * 3 x^2#

#y = uv#

#y' =u'v + uv'#

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Answer 2

Ezra Adams

To differentiate ( f(x) = e^x(x^3 - 1) ), you can use the product rule, which states that if you have two functions ( u(x) ) and ( v(x) ), then the derivative of their product is ( u'(x)v(x) + u(x)v'(x) ). Let ( u(x) = e^x ) and ( v(x) = x^3 - 1 ). Then, differentiate both functions separately and apply the product rule.

( u'(x) = e^x ) (since the derivative of ( e^x ) is itself)
( v'(x) = 3x^2 ) (since the derivative of ( x^3 - 1 ) is ( 3x^2 ))

Now, apply the product rule:
( f'(x) = u'(x)v(x) + u(x)v'(x) )
( f'(x) = e^x(x^3 - 1) + e^x(3x^2) )
( 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) ) 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.