7–14. Find the derivative the following ways:
a. Using the Product Rule (Exercises 7–10) or the Quotient Rule (Exercises 11–14). Simplify your result.
y = x² - a² / x-a, where a is a constant

Verified step by step guidance

1. Step 1: Identify the function as a quotient, where the numerator is \( x^2 - a^2 \) and the denominator is \( x - a \).
2. Step 2: Recall the Quotient Rule for derivatives, which states that if \( y = \frac{u}{v} \), then \( y' = \frac{u'v - uv'}{v^2} \).
3. Step 3: Differentiate the numerator \( u = x^2 - a^2 \) to get \( u' = 2x \), since \( a^2 \) is a constant and its derivative is zero.
4. Step 4: Differentiate the denominator \( v = x - a \) to get \( v' = 1 \), since \( a \) is a constant and its derivative is zero.
5. Step 5: Substitute \( u, u', v, \) and \( v' \) into the Quotient Rule formula to find \( y' = \frac{(2x)(x-a) - (x^2 - a^2)(1)}{(x-a)^2} \) and simplify the expression.

Key Concepts

Product Rule

The Product Rule is a formula used to find the derivative of the product of two functions. If u(x) and v(x) are two differentiable functions, the derivative of their product is given by (u*v)' = u'v + uv'. This rule is essential when dealing with expressions where two functions are multiplied together, allowing for the correct application of differentiation.

Quotient Rule

The Quotient Rule is used to differentiate a function that is the quotient of two other functions. If u(x) and v(x) are differentiable functions, the derivative of their quotient is given by (u/v)' = (u'v - uv') / v². This rule is particularly important when the function is expressed as a fraction, ensuring that the differentiation accounts for both the numerator and denominator.

Simplification of Derivatives

Simplification of derivatives involves reducing the expression obtained after differentiation to its simplest form. This may include factoring, canceling common terms, or combining like terms. Simplifying the derivative is crucial for clarity and ease of interpretation, especially when further analysis or evaluation is required.