UNLOCK YOUR CHILD'S
POTENTIAL AND CREATIVITY
WITH A FREE TRIAL CLASS
DEVELOP TECHNICAL, SOFT, &
ENTREPRENEURIAL SKILLS
AGE 7-16 YEARS
CLAIM YOUR $10 ROBLOX/AMAZON/MINECRAFT GIFT
CARD BY ATTENDING A FREE TRIAL CLASS
BOOK A FREE TRIAL
Select Your Subject of Choice

    Please enter name

    Please enter email


    Existing knowledge in the chosen stream

    *No credit card required.

    Chain Rule: Guide with Theorem and Examples

    |

    The Chain Rule is a powerful concept in calculus that provides a systematic way to find the derivatives of complex functions. In this blog post, we’ll break down the Chain Rule, introduce its fundamental theorem, and illustrate its application through easy-to-follow examples.

    The Chain Rule Theorem:

    Starting with the formal statement of the Chain Rule. If we have a composite function: 

    y = f(g(x))

    f = inner function

    g = outer function

    The Chain Rule can be expressed as follows:

    dy/dx = dy/dg x dg/dx

    In simpler terms, the rate of change of the composite function is equal to the product of the rate of change of the outer function with respect to the inner function, and the rate of change of the inner function with respect to the original variable.

    Example 1: Basic Chain Rule Application

    Let’s consider the function y = (3x2 + 2x)5 

    Here, the outer function is f(u) = u5

    and

    the inner function is g(x) = 3x2 + 2x 

    Applying the Chain Rule, we get:

    dy/dx = 5(3x2 + 2x)4 x (6x + 2) 

    Recommended Reading: What is the Derivative of sec x?

    Example 2: Chain Rule with Trigonometric Functions

    Now, let’s look at y = sin(2x). 

    The inner function is g(x) = 2x

    and

    the outer function is f(u) = sin(u). 

    Applying the Chain Rule, we get:

    dy/dx = cos(2x) x 2 

    Example 3: Chain Rule with Exponential Functions

    Consider y = e3x

    Here, the inner function is g(x) = 3x

    and

    the outer function is f(u) = eu

    Applying the Chain Rule, we get:

    dy/dx = 3e3x

    Recommended Reading: What is cos 5pi 6?

    Conclusion:

    The Chain Rule is a fundamental tool that simplifies the process of finding derivatives for composite functions. By understanding its theorem and practicing with examples, you’ll gain confidence in differentiating complex mathematical expressions. As you dive deeper into calculus, the Chain Rule will continue to be a reliable companion, enabling you to dissect intricate functions with ease.

    Moonpreneur understands the needs and demands this rapidly changing technological world is bringing with it for our kids. Our expert-designed Advanced Math course and Math Quiz for grades 3rd, 4th, 5th, and 6th will help your child develop math skills with hands-on lessons, excite them to learn, and help them build real-life applications. 

    Register for a free 60-minute Advanced Math Workshop today!

    Moonpreneur

    Moonpreneur

    Moonpreneur is an ed-tech company that imparts tech entrepreneurship to children aged 6 to 15. Its flagship offering, the Innovator Program, offers students a holistic learning experience that blends Technical Skills, Power Skills, and Entrepreneurial Skills with streams such as Robotics, Game Development, App Development, Advanced Math, Scratch Coding, and Book Writing & Publishing.
    Subscribe
    Notify of
    guest

    0 Comments
    Inline Feedbacks
    View all comments

    RELATED ARTICALS

    Explore by Category

    MOST POPULAR

    GIVE A GIFT OF $10
    MINECRAFT GIFT
    TO YOUR CHILD

    JOIN A FREE TRIAL CLASS

    FREE PRINTABLE MATH WORKSHEETS

    DOWNLOAD 3rd GRADE MATH WORKSHEET
    Download Now

    DOWNLOAD 4rd GRADE MATH WORKSHEET
    Download Now

    DOWNLOAD 5rd GRADE MATH WORKSHEET
    Download Now

    DOWNLOAD 4rd GRADE MATH WORKSHEET
    Download Now

    MATH QUIZ FOR KIDS - TEST YOUR KNOWLEDGE

    MATH QUIZ FOR GRADE 3

    Start The Quiz

    MATH QUIZ FOR GRADE 4

    Start The Quiz

    MATH QUIZ FOR GRADE 5

    Start The Quiz

    MATH QUIZ FOR GRADE 6

    Start The Quiz