Differentiation of the Product

Differentiation of the Products

Introduction

Differentiation is the process of finding the rate of change of a value, also known as the derivative. It is a subfield of calculus extensively used in physics to work out formulas and the rate of change of a specific parameter, such as speed. If you did not know, speed is a derivative of distance concerning time. So, if you try to calculate the speed, you essentially differentiate distance concerning the time

Speed = Distance/Time

Product rule

The product rule of calculus, also known as the Leibniz rule, is used to find the derivative of any given function present in a product form of two differentiable functions. 

The product rule is used to find the derivative of a function in the form of f(x).g(x), in which both the f(x) and g(x) are differentiable entities. 

Product rule definition

For the sake of defining the product rule, let us suppose that the function in question is p(x). Such that p(x) is a final product obtained after the multiplication of any two differentiable quantities f(x) and g(x). 

And therefore, the product rule is the method through which you can find the derivative of this ‘function p(x)’. 

Let’s say f(x) and g(x) are the two differentiable functions. To differentiate the product of these functions, you can differentiate f(x) and g(x), which will give you f’ and g’.

Then, the derivative of the product of f(x)*g(x) will be the sum of g(x) multiplied by f'(x) and f(x) multiplied by g'(x).

To find the derivative of a function of form f(x).g(x), you need to use the product rule formula. 

Product rule formula

The product rule, as mentioned earlier, is essential to solve the derivatives or to evaluate the differentiation of the product of any two functions. The product rule formula is: 

      d/dx × f(X)= d/dx{ u(X).v(X)}=[v(X)×u'(X)+u(X)×v'(X)]

Here, 

• f(x) is the product of the differentiable functions u(x) and v(x)
• u(x) and v(x) are the differentiable products
• u’(x) is the derivative of u(x) and
• v’(x) is the derivative of v(x)

Conclusion

This article dealt in-depth with the product rule, its formula and derivational proofs. So, from the information mentioned above, you can now easily deduce the product rule answers to any problem. Just make sure to practice as much as you can. Chapters like differentiation and integration from an exam point of view become very crucial as these chapters might seem complex at first, but once understood are very easy to score and can be proved if mastered as an added asset to your skills.