Find the derivative the following ways:
Using the Product Rule or the Quotient Rule. Simplify your result.
f(x) = (x - 1)(3x + 4)

Verified step by step guidance

1. Step 1: Identify the functions to apply the Product Rule. Here, we have two functions: \( u(x) = x - 1 \) and \( v(x) = 3x + 4 \).
2. Step 2: Recall the Product Rule formula: \( (uv)' = u'v + uv' \). This means we need to find the derivatives of \( u(x) \) and \( v(x) \).
3. Step 3: Differentiate \( u(x) = x - 1 \). The derivative \( u'(x) \) is 1, since the derivative of \( x \) is 1 and the derivative of a constant is 0.
4. Step 4: Differentiate \( v(x) = 3x + 4 \). The derivative \( v'(x) \) is 3, since the derivative of \( 3x \) is 3 and the derivative of a constant is 0.
5. Step 5: Apply the Product Rule: \( f'(x) = u'v + uv' = (1)(3x + 4) + (x - 1)(3) \). Simplify the expression to find the derivative.

Key Concepts

Product Rule

The Product Rule is a formula used to find the derivative of the product of two functions. If you have two functions, u(x) and v(x), the derivative of their product is given by f'(x) = u'v + uv'. This rule is essential when differentiating expressions where two functions are multiplied together, as it allows for the correct application of differentiation principles.

Quotient Rule

The Quotient Rule is used to differentiate a function that is the ratio of two other functions. If f(x) = u(x)/v(x), the derivative is given by f'(x) = (u'v - uv')/v^2. This rule is crucial when dealing with fractions of functions, ensuring that the differentiation accounts for both the numerator and denominator appropriately.

Simplification of Derivatives

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