Derivative formula of u v
Derivatives are a fundamental concept in calculus that represent the rate of change of a function. When dealing with the product of two functions, u and v, the derivative of their product can be calculated using the product rule.

Product Rule
The product rule states that the derivative of the product of two functions, u and v, is given by the formula:
(d/dx) [u(x) v(x)] = u'(x) v(x) + u(x) v'(x)
where u'(x) and v'(x) are the derivatives of u(x) and v(x) with respect to x, respectively.

Explanation
When differentiating the product of two functions, u and v, the derivative of the first function (u) is multiplied by the second function (v), and vice versa. This rule allows us to find the derivative of the product without explicitly expanding the product.

Example
For example, let u(x) = 2x and v(x) = x^2. To find the derivative of u(x) v(x), we first differentiate u(x) to get u'(x) = 2, and differentiate v(x) to get v'(x) = 2x. Applying the product rule, we have:
(d/dx) [2x * x^2] = 2 * x^2 + 2x * 2x = 2x^2 + 4x^2 = 6x^2
Therefore, the derivative of the product u(x) v(x) is 6x^2.
In conclusion, the derivative formula of u v involves applying the product rule, which simplifies the process of finding the derivative of the product of two functions.