The product rule of derivatives is . We can read this as the derivative of the product of two functions is equal to the derivative of the first function, multiplied with the second function as it is, plus the first function as it is, multiplied with the derivative of the second function. This product rule can be proved using the 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 the limit of both sides as , we have
NOTE: If we extended the product of three functions, then
Example: Find the derivative of
We have the given function as
Differentiating with respect to variable , we get
Now using the formula of the derivative of a square root, we have