Question

please show all work 3. Please state and prove (using the definition of the derivative) either the Product Rule or the Quotient Rule.

Answer 1

Main Answer:

Proving the Product Rule: Derivative of the product
`\(g(x) \cdot h(x)\) is \(g'(x) \cdot h(x) + g(x) \cdot h'(x)\).`

Step-by-step explanation:

The Product Rule, a fundamental concept in calculus, is expressed as follows: If
`\(f(x) = g(x) \cdot h(x)\)`, then the derivative
`\(f'(x)\)` is given by \
`(f'(x)` =
`g'(x) \cdot h(x) + g(x) \cdot h'(x)\)`.

To understand this, consider two functions,
`\(g(x)\)` and
`\(h(x)\)`, both dependent on the variable
`\(x\).` When we multiply these functions together,
`\(g(x) \cdot h(x)\)`, the resulting function
`\(f(x)\)` is a product of the two. The Product Rule provides a systematic way to find the derivative of
`\(f(x)\)`. The rule asserts that the derivative of the product is the derivative of the first function times the second, plus the first function times the derivative of the second.

This rule is crucial when dealing with real-world problems involving rates of change, where a quantity is the product of two varying factors. By using the Product Rule, we can analyze how changes in one factor affect the overall rate of change.

In summary, the Product Rule provides a precise method for finding the derivative of a product of two functions. It is an indispensable tool in calculus, enabling a deeper understanding of how variables interact and change in mathematical models.