Part 1/4:
Understanding the Product Rule in Calculus
The product rule is a fundamental theorem in calculus that allows us to calculate the derivative of the product of two functions. It states that if you have two functions, ( f(x) ) and ( g(x) ), the derivative of their product ( y = f(x) \times g(x) ) can be found using the formula:
[
\frac{dy}{dx} = f'(x) \times g(x) + f(x) \times g'(x)
]
Deriving the Product Rule
To understand the product rule more deeply, we start with two functions ( u = f(x) ) and ( v = g(x) ). When we multiply these together, they create an area represented by ( A = u \times v ).