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How do you differentiate #f(x)= (x^2+1)(x^3+1) # using the product rule?

Answer 1

Isaiah Allen

#f'(x) = 5x^4 + 3x^2 + 2x#

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

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

#f'(x) = 2x^4+2x +3x^4+3x^2#

#f'(x) = 5x^4 + 3x^2 + 2x#

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Answer 2

Ezra Adams

To differentiate ( f(x) = (x^2 + 1)(x^3 + 1) ) using the product rule, you would follow these steps:

Identify the two functions being multiplied: ( u = x^2 + 1 ) and ( v = x^3 + 1 ).
Apply the product rule: ( f'(x) = u'v + uv' ).
Find the derivatives of ( u ) and ( v ): ( u' = 2x ) and ( v' = 3x^2 ).
Substitute the derivatives and the original functions into the product rule formula: ( f'(x) = (2x)(x^3 + 1) + (x^2 + 1)(3x^2) ).
Simplify the expression: ( f'(x) = 2x^4 + 2x + 3x^4 + 3x^2 ).
Combine like terms: ( f'(x) = 5x^4 + 3x^2 + 2x ).

Therefore, the derivative of ( f(x) = (x^2 + 1)(x^3 + 1) ) using the product rule is ( f'(x) = 5x^4 + 3x^2 + 2x ).

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Answer from HIX Tutor

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