Derivatives Find and simplify the derivative of the following functions.
h(x) = (x−1)(2x²-1) / (x³-1)

Verified step by step guidance

Step 1: Identify the function as a quotient of two functions, where the numerator is $(x-1)(2x^2-1)$ and the denominator is $x^3-1$. We will use the quotient rule for derivatives, which is given by $\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \cdot u' - u \cdot v'}{v^2}$, where $u$ is the numerator and $v$ is the denominator.

Step 2: Differentiate the numerator $u = (x-1)(2x^2-1)$ using the product rule. The product rule states that $\frac{d}{dx}(fg) = f'g + fg'$. Let $f(x) = x-1$ and $g(x) = 2x^2-1$. Find $f'(x)$ and $g'(x)$, then apply the product rule.

Step 3: Differentiate the denominator $v = x^3-1$. The derivative of $v$ is $v' = 3x^2$.

Step 4: Substitute $u$, $u'$, $v$, and $v'$ into the quotient rule formula. Simplify the expression obtained from the quotient rule.

Step 5: Simplify the resulting expression further, if possible, by combining like terms and reducing any common factors in the numerator and the denominator.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Derivatives

A derivative represents the rate at which a function changes at a given point. It is defined as the limit of the average rate of change of the function as the interval approaches zero. In calculus, derivatives are fundamental for understanding the behavior of functions, including their slopes and rates of change.

Product Rule

The Product Rule is a formula used to find the derivative of the product of two functions. It states that if you have two functions, u(x) and v(x), the derivative of their product is given by u'v + uv'. This rule is essential when differentiating functions that are multiplied together, as seen in the function h(x) provided.

Quotient Rule

The Quotient Rule is used to differentiate functions that are expressed as the ratio of two other functions. If h(x) = u(x)/v(x), the derivative is given by (u'v - uv')/v². This rule is crucial for the function h(x) in the question, as it involves a division of two polynomial expressions.

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