How do you differentiate #g(x) =x (4x^6 + 5) # using the product rule?

Answer 1

Scarlett Allison

#28x^6+5#

According to the product rule, if

#g(x)=a(x)*b(x)#

then

#g'(x)=a'(x)b(x)+a(x)b'(x)#

Substituting in from the question, #a(x)=x# and #b(x)=4x^6+5#, then

#a'(x)=1#

#b'(x)=24x^5#

We can now locate that

#g'(x)=1(4x^6+5)+x(24x^6)#

and when the brackets are expanded,

#=28x^6+5#

By distributing the function and simply applying the power rule, you can verify the following answer:

#g(x)=4x^7+5x#

#g'(x)=28x^6+5#

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Answer 2

Isabella Ackerman

To differentiate ( g(x) = x(4x^6 + 5) ) using the product rule, you would apply the formula: [ (uv)' = u'v + uv' ] where ( u = x ) and ( v = 4x^6 + 5 ). Then, differentiate ( u ) and ( v ) individually. The derivative of ( u ) is ( u' = 1 ), and the derivative of ( v ) is ( v' = 24x^5 ). Finally, apply the product rule formula to find ( g'(x) ).

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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.