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

Answer 1

Scarlett Allison

#f'(x) = frac{2(3-x)}{(x+3)^{4}}#

#f(x) = (x-1) (x+3)^{-3}#

#f'(x) = frac{d}{dx}( (x-1) (x+3)^{-3} )#

#= (x+3)^{-3} frac{d}{dx}( x-1 ) + (x-1) frac{d}{dx}( (x+3)^{-3} )#

#= (x+3)^{-3} ( 1 ) + (x-1) ( (-3) (x+3)^{-4} (1) )#

#= (x+3)^{-4} ( (x+3) - 3 (x-1) ) #

#= frac{2(3-x)}{(x+3)^{4}}#

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Answer 2

Samantha Adkison

To differentiate ( f(x) = (x - 1)\left(\frac{1}{(x + 3)^3}\right) ) using the product rule, you would follow these steps:

Identify the two functions being multiplied: ( u(x) = x - 1 ) and ( v(x) = \frac{1}{(x + 3)^3} ).
Calculate the derivatives of each function: ( u'(x) = 1 ) and ( v'(x) = -3\frac{1}{(x + 3)^4} ).
Apply the product rule formula: ( (uv)' = u'v + uv' ).
Substitute the values obtained in steps 2 and 3 into the product rule formula.

So, the derivative of ( f(x) ) using the product rule is:

[ f'(x) = (x - 1)\left(-3\frac{1}{(x + 3)^4}\right) + \left(1\right)\left(\frac{1}{(x + 3)^3}\right) ]

[ f'(x) = -3\frac{x - 1}{(x + 3)^4} + \frac{1}{(x + 3)^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.