chain rule examples

chain rule examples When to Use the Chain Rule The chain rule is used to differentiate any composite function of the form y f g That is a function that has an inner function with an outer function applied to it For example y 3 2 5 is made of the functions g 3 2 and f 5

The chain rule tells us how to find the derivative of a composite function Brush up on your knowledge of composite functions and learn how to apply the chain rule correctly The chain rule says d d x f g x f g x g x It tells us how to differentiate composite functions Example What is ddx 5x 2 3 The Chain Rule says the derivative of f g x f g x g x 5x 2 3 is made up of g 3 and 5x 2 f g g 3 g x 5x 2 The individual derivatives are f g 3g 2 by the Power Rule g x 5 So ddx 5x 2 3 3g x 2 5 15 5x 2 2

chain rule examples

chain rule examples
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Example 1 Use the Chain Rule to differentiate R left z right sqrt 5z 8 Show Solution We ve already identified the two functions that we needed for the composition but let s write them back Section 3 9 Chain Rule For problems 1 27 differentiate the given function f left x right left 6 x 2 7x right 4 Solution g left t right left 4 t 2 3t 2 right 2 Solution y sqrt 3 1 8z Solution R left w right csc left 7w right Solution

This formula allows us to derive a composition of functions such as but not limited to f g x Here we will look at a summary of the chain rule Additionally we will explore several examples with answers to understand the Example PageIndex 3 Using the Generalized Chain Rule Calculate w u and w v using the following functions begin align w f x y z 3x 2 2xy 4z 2 4pt x x u v e u sin v 4pt y y u v e u cos v 4pt z z u v e u end align Solution The formulas for w u and w v are

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The chain rule states that the derivative of f g x is f g x g x In other words it helps us differentiate composite functions For example sin x is a composite function because it can be constructed as f g x for f x sin x and g x x Worked example Chain rule with table Google Classroom About Transcript Through a worked example we explore the Chain rule with a table Using specific x values for functions f and g and their derivatives we collaboratively evaluate the derivative of a composite function F x f g x

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chain rule examples - Example 1 Use the Chain Rule to differentiate R left z right sqrt 5z 8 Show Solution We ve already identified the two functions that we needed for the composition but let s write them back