Using the chain rule...?

Question

How can i tell what the inside and outside functions are when using the chain rule to take derivatives?

Answer 1

Think about your order of operations. If you were evaluting the function for some number, what would you do first? That function is your *inside* function.

Answer 2

It's only for functions of the form f(g(x))

For example, with 1/(x^3), the inside function is x^3 and the outside function is 1/x.

The inside function is the one you "put inside" the other one.

Answer 3

In intuitive terms, if a variable, y, depends on a second variable, u, which in turn depends on a third variable, x, then the rate of change of y with respect to x can be computed as the product of the rate of change of y with respect to u multiplied by the rate of change of u with respect to x.


In integration, the counterpart to the chain rule is the substitution rule.

Example
Suppose, for example, that one is climbing a mountain at a rate of 0.5 kilometres per hour. The temperature is lower at higher elevations; suppose the rate by which it decreases is 6 °C per kilometre. If one multiplies 6 °C per kilometre by 0.5 kilometre per hour, one obtains 3 °C per hour. This calculation is a typical chain rule application.

Answer 4

it depends but it is clear in most cases like fro example
(sinx)^1/2

now in this f(y)=y^1/2
and g(x)=sinx
and this function is f(g(x))
its easy.