The derivative of a function of a real variable in mathematics measures the sensitivity of the function value to a change in its argument. Calculus is heavily reliant on derivatives. The chain rule is the method used to find the derivative of a composite function (e.g., cos 2x, log 2x, etc.). It’s also known as the composite function rule. The chain rule only applies to composite functions. So, before we begin the chain rule formula, let us first define the composite function and how it can be differentiated.
To differentiate a function in Calculus, the product rule is used. The product rule is applied when a given function is the product of two or more functions. If the problems are a combination of two or more functions, the derivatives of those functions can be found using the Product Rule.
In this article, we will learn about the different ways in which we can differentiate functions.
What is the Product Rule in Calculus?
In calculus, the product rule is a method for determining the derivative of any function given in the form of a product obtained by multiplying any two differentiable functions. In other words, the derivative of a product of two differentiable functions equals the sum of the product of the second function with differentiation of the first function and the product of the first function with differentiation of the second function.
When Do You Need to Use it & Product Rule in Differentiation?
These are two extremely useful rules for distinguishing functions. In general, we use the chain rule to differentiate a ‘function of a function,’ such as f(g(x)). When differentiating two functions multiplied together, such as f(x)g(x), we use the product rule.
Consider the expression f(x) = sin (3x). This is an example of a ‘composite’ function, which is essentially a ‘function of a function.’ In this example, the two functions are as follows: Function one multiplies x by three; function two takes the sine of the answer provided by function one. To distinguish between these types of functions, we must employ the chain rule.
What Exactly Is The Chain Rule?
This chain rule is also referred to as the outside-in rule, the composite function rule, or the function of a function rule. It is only used to compute the derivatives of the composite functions.
The Chain Rule Theorem: Let f be a real-valued function that is a combination of two other functions, g and h. That is, f = g o h. If u = h(x) and du/dx and dg/du exist, then this can be expressed as: change in f/change in x = change in g /change in u change in u /change in x
This is expressed in Leibniz notation as an equation, df/dx = dg/du.du/dx.
Steps in The Chain Rule
Step 1: Determine The Chain Rule: The function must be a composite function, which means it is nested over another.
Step 2: Determine the inner and outer functions.
Step 3: Leaving the inner function alone, find the derivative of the outer function.
Step 4: Determine the inner function’s derivative.
Step 5: Add the results from steps 4 and 5.
Step 6: Reduce the complexity of the chain rule derivative.
Chain Rule Applications
This chain rule has numerous applications in physics, chemistry, and engineering. We use the chain rule:
- To calculate the pressure’s time rate of change.
- In order to compute the rate of change of distance between two moving objects,
- To determine the position of an object that moves to the right and left in a specific interval.
- To determine whether or not a function is increasing or decreasing
- To calculate the average molecular speed’s rate of change
If you want to learn more about differentiation, as well as the different methods of differentiation you can visit the Cuemath website. Here, you will be able to understand the concepts in a fun way!