Which rule is used to differentiate a product of two functions?

The rule used to differentiate a product of two functions is the Product Rule. This rule states that if you have two functions, let's call them ( u(x) ) and ( v(x) ), the derivative of their product ( u(x)v(x) ) is given by the formula:

[

\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)

]

This means you take the derivative of the first function, multiply it by the second function, and then add it to the first function multiplied by the derivative of the second function. This approach is necessary because the derivative of a product involves contributions from both functions, reflecting the interplay of their rates of change.

Other differentiation rules, such as the Quotient Rule, the Chain Rule, and the Power Rule, serve different purposes and are applicable in other contexts. For instance, the Quotient Rule is used for differentiating the ratio of two functions, the Chain Rule is relevant for composite functions, and the Power Rule applies to functions in the form of ( x^n ). These rules do not pertain to products like the Product Rule does, which is why the Product Rule is the correct choice for differenti