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Proving the chain rule

Proving the chain rule for derivatives.
The chain rule tells us how to find the derivative of a composite function:
The AP Calculus course doesn't require knowing the proof of this rule, but we believe that as long as a proof is accessible, there's always something to learn from it. In general, it's always good to require some kind of proof or justification for the theorems you learn.

First, we would like to prove two smaller claims that we are going to use in our proof of the chain rule.

(Claims that are used within a proof are often called lemmas.)

1. If a function is differentiable, then it is also continuous.

Khan Academy video wrapper
Proof: Differentiability implies continuitySee video transcript

2. If function u is continuous at x, then Δu0 as Δx0.

Khan Academy video wrapper
If function u is continuous at x, then Δu→0 as Δx→0 See video transcript

Now we are ready to prove the chain rule!

Khan Academy video wrapper
Chain rule proofSee video transcript

Bonus: We can use the chain rule and the product rule to prove the quotient rule.

The quotient rule tells us how to find the derivative of a quotient:
Khan Academy video wrapper
Quotient rule from product & chain rulesSee video transcript

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