The Product Rule is used to derive functions of the form \( y= u*v \). This form occurs when you must derive two functions which are multiplied by each other. When using The Product Rule it is not important which function of x is assigned to which of the letters. The Product Rule states:
\[ If y(x) = uv then \frac {dy}{dx} = uv' + vu' \]
Some examples of this are provided below:
Example One: Find \( \frac {dy}{dx} if y(x) = x^3 * sin(x)
\( Let: \x^3 = u \sin(x) = v \) \( \therefore 3x^2 = u' \cos(x) = v' \)
The Product Rule: \( If y(x) = uv then \frac {dy}{dx} = uv' + vu' \)
Substituting our above values into the equation yields: