Chain rule in calculus

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Before discussing what is the chain rule in calculus, let’s first talk about calculus. The infinitesimal calculus or calculus refers to a mathematical study that includes the learning of continuous change. Isaac Newton and Gottfried Wilhelm Leibniz developed the concept of modern calculus in the 17th century. The chain rule is just a formula of calculus and, in this article, we will be discussing that. So, let’s get started.

What is the chain rule in calculus?

The chain rule refers to a formula in calculus that one uses to express the derivative of the composite of either two or more than two functions. Composite function refers to those functions where one function is inside another. An individual can express the chain rule in two different notations: Lagrange’s notation and Leibniz’s notation. 

For the calculation of the chain rule, one can assume that there are two differentiable functions, f(x) and g(x). Their composite function therefore will be f(g(x)) for the value of x. For this, one needs to calculate the value of g(x) initially. 

Then, evaluate the function f at the calculated value of g(x). Doing this will chain the results together and hence give you the composition of those two functions.

The formula of the chain rule 

The formula of the chain rule is stated as

dy/dx = dy/du.du/dx

Here, dy/dx is the derivative of y with respect to x

dy/du is the derivative of y with respect to u

du/dx is the derivative of u with respect to x.

In simpler language, if a variable y is dependent on another variable u, and that variable is again dependent on a variable x, then that means variable y is dependent on variable x. This is Leibniz’s notation.

For example, let’s assume that

p=f(q); therefore, p is a function of q.

Now, p=f(r), which means p is a function of r as well. 

r=f(q), hence, r is a function of q. 

So then, dp/dq = dp/dr.dr/dq

Thus, using the chain rule makes the difficult differentiation process much easier and simpler. 

There are various examples of problems available that one can solve by using this formula of the chain rule. It provides us with a way to easily evaluate the derivative of a composite function. 

History of the chain rule

Gottfried Wilhelm Leibniz is the first to use the formula of the chain rule. In a memoir of 1676, the mention of this rule was made by him. The use of this rule was also done by a French mathematician, Guillaume de l’Hôpital, who used it implicitly in a textbook of his.

The applications of the chain rule

The application of chain rule is distinctively visible in the fields of physics, engineering, and chemistry. Many disciples also use the chain rule for studying related rates. The chain rule is possible for instances where the nested functions have more than one variable to depend on. 

In physics, it is often seen that a single physical quantity is dependent on another physical quantity and that quantity is dependent on another. As a result, the chain rule in physics has broad application. For example, the uses it is seen in simple harmonic motion, electromagnetic induction, and kinematics.

In chemistry, the same reason follows as well. Various equations in chemistry state the dependency of one physical quantity on another, which has a dependency on another. For example, the ideal gas law, kinetic theory of gases use the chain rule. 

So, after reading this article, you must answer all the questions regarding what is the chain rule in calculus or the formula of the chain rule. This is for those first-time learners who need a basic idea regarding the chain rule in calculus.