Chain rule

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The chain rule is a rule of derivation that tells us that, having a variable y that depends on u, and if it depends on the variable x, then the rate of change of y with respect to x can be estimated as the product of the derivative of y with respect to u by the derivative of u with respect to x.

In mathematical terms, it can be translated like this:

To use this rule well, it is important to be able to correctly identify whether a function is composite, as well as to determine the exterior and interior function.

For example, if we have (4x + 7) 2, it is a compound function where 4x + 7 is the internal function to which we can assign the name y, while the external function is y2.

This rule is useful, for example, in trigonometric functions that affect polynomials or algebraic expressions, as we will see in the examples later.

Chain rule examples

We will see some examples of application of the chain rule:

Now, a second example with a trigonometric function:

Finally, a more complex example of a squared trigonometric function:

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