# How do you prove the chain rule?

see below

i'm gonna make this up on the spot so mea culpa if it's messy but I find Taylor expansions really useful in these sorts of situations

and by definition we have

Using Taylor, we're gonna expand the first bit as follows:

By signing up, you agree to our Terms of Service and Privacy Policy

To prove the chain rule, start with the definition of the derivative. Then, apply the limit definition of the derivative to the composition of two functions, ( f(g(x)) ). Using algebraic manipulation and the properties of limits, simplify the expression until it matches the form of the chain rule. Finally, show that as ( h ) approaches zero, the expression converges to the chain rule formula, which states that the derivative of ( f(g(x)) ) with respect to ( x ) is equal to the derivative of ( f ) with respect to ( g ) multiplied by the derivative of ( g ) with respect to ( x ).

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7