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How do you differentiate #g(x) = (x-7)(x^2+3)# using the product rule?

Answer 1

Aurora Alvarado

#3*x^2-14*x+3#

We have#g(x) = (x-7)(x^2+3)# applying product rule we get

using( #dy/dx=x^n=>n*x^(n-1)#

#dy/dx=k(constant)=>0 #) #=>2*x^2-14*x+x^2+3#

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Answer 2

Aurora Alvarado

To differentiate g(x) = (x - 7)(x^2 + 3) using the product rule:

Identify the functions u(x) and v(x).
- Let u(x) = x - 7
- Let v(x) = x^2 + 3
Apply the product rule: (u(x)v(x))' = u'(x)v(x) + u(x)v'(x)
- Differentiate u(x) to get u'(x) = 1
- Differentiate v(x) to get v'(x) = 2x
Substitute the derivatives and original functions into the product rule formula:
- g'(x) = (1)(x^2 + 3) + (x - 7)(2x) = x^2 + 3 + 2x(x - 7)
Simplify the expression:
- g'(x) = x^2 + 3 + 2x^2 - 14x = 3x^2 - 14x + 3

Therefore, the derivative of g(x) = (x - 7)(x^2 + 3) using the product rule is g'(x) = 3x^2 - 14x + 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.