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Calculus – Chain Rule




 
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The Chain Rule is used to find the derivative of composite functions.

The Chain Rule

If f and g are both differentiable and F(x) is the composite function defined by F(x) = f(g(x)) then F is differentiable and F ’ is given by the product

F ’(x) = f ‘(g(x)) g’(x)

In Leibniz notation, if y = f(u) and u = g(x) are both differentiable functions, then

Note: In the Chain Rule, we work from the outside to the inside. We differentiate the outer function and then we multiply with the derivative of the inner function.

Example:

Find the derivatives of each of the following

Solution:

Example:

Differentiate y = (2x + 1)5(x3x +1)4

Solution:

In this example, we use the Product Rule before using the Chain Rule.




Videos

Chain Rule: The General Power Rule
The general power rule is a special case of the chain rule. It is useful when finding the derivative of a function that is raised to the nth power. The general power rule states that this derivative is n times the function raised to the (n-1)th power times the derivative of the function.
This tutorial presents the chain rule and a specialized version called the generalized power rule. Several examples are demonstrated.
Chain Rule: The General Exponential Rule
The exponential rule is a special case of the chain rule. It is useful when finding the derivative of e raised to the power of a function. The exponential rule states that this derivative is e to the power of the function times the derivative of the function.
Derivatives of Exponential Functions. Just some examples of finding derivatives of functions involving exponentials.


 
Chain Rule: The General Logarithm Rule
The logarithm rule is a special case of the chain rule. It is useful when finding the derivative of the natural logarithm of a function. The logarithm rule states that this derivative is 1 divided by the function times the derivative of the function.
Examples using the Chain Rule.


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