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

Question

Christopher Agresta · Accepted Answer

#g'(x)=4x^3-6x^2+2x-2# #"Given " g(x)=f(x).h(x)" then"# #color(red)(bar(ul(|color(white)(2/2)color(black)(g'(x)=f(x)h'(x)+h(x)f'(x))color(white)(2/2)|)))larr" product rule"# #"here " f(x)=x^2+1rArrf'(x)=2x# #"and " h(x)=x^2-2xrArrh'(x)=2x-2# #rArrg'(x)=(x^2+1)(2x-2)+(x^2-2x).2x# #=2x^3-2x^2+2x-2+2x^3-4x^2# #=4x^3-6x^2+2x-2#

Scarlett Allison · Answer

undefined To use the product rule to differentiate $ g(x) = (x^2 + 1)(x^2 - 2x) $, follow these steps:

1. Identify the functions $ f(x) = x^2 + 1 $ and $ h(x) = x^2 - 2x $.
2. Apply the product rule, which states that the derivative of the product of two functions is the derivative of the first function times the second function, plus the first function times the derivative of the second function.
3. Calculate the derivatives of $ f(x) $ and $ h(x) $.
4. Substitute the derivatives and the original functions into the product rule formula.
5. Simplify the expression to get the derivative of $ g(x) $.

Using the product rule, the derivative of $ g(x) $ is:
$$ g'(x) = (2x)(x^2 - 2x) + (x^2 + 1)(2x - 2) $$

After simplifying, the final result is:
$$ g'(x) = 2x^3 - 4x^2 + 2x - 2 $$