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April 8, 2025Derivatives of Composite and Implicit Functions: Differentiation is the process of finding the derivative, or rate of change, of a function. In general, the derivative of algebraic functions can easily be found using the standard differentiation formulas. So, what will happen if one function is inside the other function? These are called composite functions. How can we find its derivative? What kind of methods are we going to use to find the derivative of composite functions? In this article, let us learn to find the derivatives of two types of functions composite and implicit functions with the help of some standard differentiation formula.
If
Some examples of composite functions are:
As we know, derivatives are an essential part of calculus. They help us calculate the rate of change, maxima and minima of functions. Derivatives, by definition, are given using limits, which is called the first principle of derivatives. We know to find the derivatives of some standard functions, but in some cases, we need to deal with complex functions composed of two or more functions. So, it is hard to calculate the derivatives of such functions directly.
Learn about Algebra of Derivatives of Functions Here
To determine the derivative of the composite function, we differentiate the first function with respect to the second function and then differentiate the second function with respect to the variable.
If
The composite function derivatives are determined using the chain rule, also known as the composite function rule.
Chain Rule
Let
Suppose
Derivative of
Another way of writing the derivative of composite functions using the chain rule is:
derivative of
In simple words, to find the derivative of a composite function, find the derivative of the outside function with respect to the inside function. Then, find the derivative of the inside function with respect to the variable, and multiply them. The result is the derivative of the composite function.
Let us see the proof for the chain rule below,
If
Now,
Therefore,
As
Thus,
The derivative of the composite function in one variable is determined using the chain rule formula. Let us see an example.
Example: Find the derivative of the following composite function:
Solution: Given that,
As we know, the composite functions can be determined by using chain rule, so
Now, differentiate with respect to
Therefore,
Before looking at the implicit function, let us understand about explicit function. In general simple linear equations,
An implicit function is a function, which written in terms of both dependent and independent variables, for example,
The implicit function is of the form
The following are some of the important properties of the implicit function, which help understand the implicit function.
Property 1: The implicit function cannot be expressed in the form of
Property 2: The implicit function is always represented as a combination of variables as
Property 3: The implicit function is a non-linear function with many variables.
Property 4: The implicit function is written in terms of the dependent and independent variables.
Property 5: The vertical line drawn through the graph of an implicit function cuts it across more than one point.
As we studied, the differentiation of functions involving a single variable can easily be calculated, but the differentiation of functions involving many variables is difficult to calculate.
The differentiation of implicit function can be determined in three simple steps.
Step 1: Differentiate the entire expression
Step 2: Find
Step 3: Write the final answer, which may have both variables i.e., dependent and independent.
Let us see one example on how to find the derivative of an implicit function
Example: Find
Solution: Given that,
Let us differentiate the given expression with respect to
Q.1. Find the derivative of the composite function
Ans: Given:
As we know, the composite functions can be determined by using chain rule, so
Now, differentiate with respect to
Therefore,
Q.2. Find the derivative of the composite function
Ans: Given:
As we know, the composite functions can be determined by using chain rule,
Now, differentiate with respect to
Therefore,
Q.3. Find the derivative of the composite function
Ans: Given:
Using the chain rule,
Therefore, we get:
Q.4. Find
Ans: Given:
Since the above equation is in the form of
Q.5. Find
Ans: Given:
Since the given function is an implicit function, we can differentiate using the chain rule,
Suppose ? and ? are two functions defined by
Important Questions on Derivatives of Composite and Implicit Functions
Q.1. What is the derivative of composite functions?
Ans: The derivative of a composite function is the product of the derivative of the outside function with respect to the inside function and the derivative of the inside function with respect to the variable. The derivatives of composite functions are determined by using the chain rule. For example, if the composite function
Q.2. What is the derivative of an implicit function?
Ans: In implicit differentiation, we differentiate each side of the equation with two variables x and y by treating one of the variables as a function of the other. This can be done by using chain rule, along with some standard formulae.
Q.3. How do you use the chain rule to find the derivative of a composite function?
Ans: The derivative of composite functions using the chain rule is derivative of
Q.4. What are implicit functions and explicit functions?
Ans: An implicit function is a function of the form
For example: The expressions
Q.5. What is
Ans: The symbol
Learn about Differentiation Formulas here
Hope this detailed article on Derivatives of Composite and Implicit Functions helps you in your preparation. In case of any query, reach out to us in the comment section and we will get back to you at the earliest.
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