Product rule of derivative is . In words we can read as derivative of quotient of two functions is equal to second function as it is and derivative of first function minus first function as it is and derivative of second function divided by square of second function. This product rule can be proving using first principle or derivative by definition.
Consider a function of the form .
First we take the increment or small change in the function.
Putting the value of function in the above equation, we get
Subtracting and adding on the right hand side, we have
Dividing both sides by , we get
Taking limit of both sides as , we have
Example: Find the derivative of
We have the given function as
Differentiation with respect to variable , we get
Now using the quotient rule of derivative, we have