Find y^'( a b by applying the Product Rule and (b) by...

Find $y^{\prime}($ a b by applying the Product Rule and (b) by multiplying the factors to produce a sum of simpler terms to differentiate.
\begin{equation}
y=\left(x^{2}+1\right)\left(x+5+\frac{1}{x}\right)
\end{equation}

Key Concepts

Differentiation of Rational Functions

Differentiation of rational functions often involves rewriting terms like 1/x as x^(-1) so that the power rule can be applied. This technique simplifies the differentiation process by converting rational expressions into forms that are compatible with the polynomial differentiation rules.

Power Rule

The power rule is a basic differentiation rule stating that the derivative of x raised to a power n is n multiplied by x raised to the power of n-1. This rule is essential when differentiating polynomial functions and is used after algebraic expansion of the product to find the derivative term by term.

Algebraic Expansion

Algebraic expansion involves multiplying out the factors of a given function to rewrite it as a sum of simpler terms. This process simplifies differentiation because once expanded, each term can be differentiated using basic rules, such as the power rule, without needing to apply the product rule.

Product Rule

The product rule is a fundamental differentiation technique used when finding the derivative of a product of two functions. According to this rule, if you have two functions f(x) and g(x), their derivative is given by f'(x)g(x) + f(x)g'(x). This method allows you to systematically differentiate the product without expanding it first.

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