Question

Use the Product Rule to differentiate the function. $$ g(x)=\left(x^{2}+1\right)\left(x^{2}-2 x\right) $$

Short answer

\( g'(x) = 4x^{3}-4x^{2}-2x-2 \)

Step-by-step solution

Identify The Component Functions

The two functions here are \( f(x) = x^{2}+1 \) and \( h(x) = x^{2}-2x \). These are the functions to be differentiated separately.

Differentiate The Component Functions

Next, the derivatives of each component function need to be found. The derivative of \( f(x) \) is \( f'(x) = 2x \) and the derivative of \( h(x) \) is \( h'(x) = 2x-2 \). The power rule is used here, which states that the derivative of \( x^n \) is \( nx^{n-1} \). Thus, for both \( x^2 \) terms, the derivative is \( 2x \), and for the \( -2x \) term, the derivative is -2 as the exponent is 1.

Apply The Product Rule

The Product Rule, \( (f \cdot h)' = f' \cdot h + f \cdot h' \), can now be applied to differentiate \( g(x) \). Substituting the derivatives and original functions into the rule gives \( g'(x) = (2x)(x^{2}-2x) + (x^{2}+1)(2x-2) \).

Simplify The Expression

The final differentiation result can be simplified by distributing and combining like terms. This leads to \( g'(x) = 2x^{3}-4x^{2} + 2x^{3}-2x-2 \), which simplifies further to \( g'(x) = 4x^{3}-4x^{2}-2x-2 \).