# The Product Rule of Derivatives

The product rule of derivatives is $\frac{d}{{dx}}\left[ {f\left( x \right)g\left( x \right)} \right] = f'\left( x \right)g\left( x \right) + f\left( x \right)g'\left( x \right)$. 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 $y = f\left( x \right)g\left( x \right)$.

First we take the increment or small change in the function.

Putting the value of function $y = f\left( x \right)g\left( x \right)$ in the above equation, we get

Subtracting and adding $f\left( x \right)g\left( {x + \Delta x} \right)$ on the right hand side, we have

Dividing both sides by $\Delta x$, we get

Taking the limit of both sides as $\Delta x \to 0$, we have

NOTE: If we extended the product of three functions, then

Example: Find the derivative of $y = \left( {2{x^2} + 5} \right)\left( {4x - 1} \right)$

We have the given function as

Differentiating with respect to variable $x$, we get

Now using the formula of the derivative of a square root, we have