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How do you differentiate #f(x)= (4x^2+5)*e^(x^2)# using the product rule?

Answer 1

Emilia Aguilar

#f'(x)=2xe^(x^2)(4x^2+9)#

Product rule: #f'(x)=u'v+v'u#

#f(x)=(4x^2+5)*e^(x^2)#

Let #u=4x^2+5# and #v=e^(x^2)#

#u'=8x# #v'=2xe^(x^2)#

#:.f'(x)=8x*e^(x^2)+2xe^(x^2)*(4x^2+5)#

#=2xe^(x^2)(4+4x^2+5)#

#=2xe^(x^2)(4x^2+9)#

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Answer 2

Aurora Alvarado

To differentiate ( f(x) = (4x^2 + 5) \cdot e^{x^2} ) using the product rule, you would apply the formula ( (uv)' = u'v + uv' ).First, let ( u = 4x^2 + 5 ) and ( v = e^{x^2} ).

Next, find the derivatives of ( u ) and ( v ):

( u' = 8x )
( v' = 2xe^{x^2} )

Finally, apply the product rule: [ f'(x) = u'v + uv' = (8x)(e^{x^2}) + (4x^2 + 5)(2xe^{x^2}) ]

So, the derivative of ( f(x) = (4x^2 + 5) \cdot e^{x^2} ) using the product rule is: [ f'(x) = 8xe^{x^2} + (4x^2 + 5)(2xe^{x^2}) ]

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Answer from HIX Tutor

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