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How do you differentiate #g(x) = (5x-2)(x^2+1)# using the product rule?

Answer 1

Paisley Johnson

# g'(x) = 2x(5x - 2 ) + 5(x^2 + 1 )#

applying the product rule to differentiate

If f(x).h(x) = g(x)

g'(x) would then equal f(x).h'(x) plus h(x).f'(x)................(A)

here f(x) =(5x - 2 ) and f'(x) = 5 #color(black)("-------------------------------------")#

h(x) # = (x^2 + 1) # hence h'(x) = 2x #color(black)("--------------------------------")#

putting these values in place of (A)

g'(x) = (5x - 2).2x + # (x^2 + 1 ).5#

# rArr g'(x) = 2x(5x - 2) + 5(x^2 + 1 )#

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Answer 2

Charles Arocha

To differentiate ( g(x) = (5x-2)(x^2+1) ) using the product rule, you apply the formula ( (uv)' = u'v + uv' ). Let ( u = 5x - 2 ) and ( v = x^2 + 1 ). Then:

( u' = 5 ) (derivative of ( 5x - 2 ) with respect to ( x ))

( v' = 2x ) (derivative of ( x^2 + 1 ) with respect to ( x ))

Now, applying the product rule:

( g'(x) = (5)(x^2 + 1) + (5x - 2)(2x) )

( g'(x) = 5x^2 + 5 + 10x^2 - 4x )

( g'(x) = 15x^2 - 4x + 5 )

So, the derivative of ( g(x) ) with respect to ( x ) is ( 15x^2 - 4x + 5 ).

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