Product Rule Formula Using the Chain Rule

Calculus employs the product rule to differentiate functions. The product rule is employed when a given function is the product of two or more other functions. If the issues are a combination of two or more functions, the Product Rule can be used to find their derivatives.Simply put, the term “product” refers to the combination of two functions that are multiplied together.

This rule, which was discovered by Gottfried Leibniz, allows us to calculate derivatives that we don’t want (or are unable to) multiply rapidly.

With another way of putting it, the product rule allows us to find the derivative of two differentiable functions that are multiplied together by combining our knowledge of both the power rule for derivatives and the sum and difference rule for derivatives.

It can be expressed as follows in simple terms: The second times the derivative of the first multiplied by the derivative of the first multiplied by the second times its own derivative equals the derivative of the second times its own derivative.

As a specific case of the chain rule, we can use it to get the calculus formula for the product rule we need. Make that f(x) can be differentiated, and the result is that h(x) equals f(x) .g (x).

ddx(f.g)=[(fg)/f][df/dx]+[(fg)/g][dg/dx]=g(df/dx)+f(dg/dx)

Hence proved.

Using chain rule and product rule together : 

The chain rule and the product rule can both be applied on the same derivative at the same time. We can see by now that these derivative rules are frequently employed in conjunction with one another. We’ve seen the power rule used in conjunction with both the product rule and the quotient rule, as well as the chain rule used in conjunction with the power rule.By combining simpler functions in one or more of the following methods, it is possible to generate several more complex functions from them:

Addition and subtraction: u(x) +v(x) and u(x)-v(x),

Multiplication and division: u(x)v(x) and u(x)v(x)

Composition: u(v(x)).

We should remember that there exist rules for differentiating functions that are produced in this manner, which is a good thing. The linearity of the derivative may be used for addition and subtraction; the product rule and the quotient rule can be used for multiplication and division; and the chain rule can be used for composition to simplify the process. Let’s take a look at these regulations again.

This means that, in addition to utilising these principles individually, it is also feasible to use them in conjunction with one another, which allows us to differentiate any combination of elementary functions. To be sure, this is not always a simple exercise, and it can be difficult at times to determine which rules should be applied and in what order, as well as whether there are any algebraic simplifications that will make the process easier to complete. To further understand the abilities required to navigate this landscape, we will look at a variety of cases in this explainer.

Examine a difficult function that incorporates many different processes, and how we might approach the differentiation by dividing it into smaller components to make it easier to understand. Consider the following scenario:

In other words, because g(x) is a collection of functions, we may use the chain rule to assist us distinguish between them. This, of course, necessitates the computation of the derivative of u(x), but even this task can be split down into smaller components. Continuing in this manner, we can continue to reduce layers of complexity from the function until we reach elementary expressions that we are familiar with and can discriminate. The following is a visual representation of what we mean.

Take note that all of the functions at the bottom of the tree are functions that we can clearly distinguish from one another. By using the necessary principles at each level, it is possible to find the derivatives of even the most difficult functions, as seen above.

We will now look at some examples in which we use this method, albeit to cases that are a little more straightforward. To begin, let us examine a factored high-degree polynomial function that has been factored several times.

Proof of chain rule : 

Conclusion : 

Calculus employs the product rule to differentiate functions. The product rule is employed when a given function is the product of two or more other functions. If the issues are a combination of two or more functions, the Product Rule can be used to find their derivatives.Simply put, the term “product” refers to the combination of two functions that are multiplied together.The chain rule and the product rule can both be applied on the same derivative at the same time. We can see by now that these derivative rules are frequently employed in conjunction with one another.