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

Answer 1

Scarlett Allison

# f'(x) = 4x^3 - 3 #

Rule of product: if f(x) = g(x).h(x)

subsequently f'(x) = g(x).h'(x) + h(x).g'(x)

using the #color(blue)(" product rule ")#

# f'(x) = x d/dx(x^3 - 3 ) + (x^3 - 3) d/dx (x) #

# = x (3x^2) + (x^3 - 3 ).1#

#rArr f'(x) = 3x^3 + x^3 - 3 = 4x^3 - 3 #

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Answer 2

Ezekiel Alexander

To differentiate ( f(x) = x(x^3 - 3) ) using the product rule, we apply the formula:

[ (uv)' = u'v + uv' ]

where ( u = x ) and ( v = x^3 - 3 ).

( u' = 1 ) (derivative of ( x ) with respect to ( x ))
( v' = 3x^2 ) (derivative of ( x^3 - 3 ) with respect to ( x ))

Applying the product rule:

[ f'(x) = (x)(3x^2) + (x^3 - 3)(1) ]

[ f'(x) = 3x^3 + x - 3 ]

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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.