University Mathematics Handbook (2015)

VII. Differential Calculus of Multivariable Functions

Chapter 6. Chain Rule

a.  If function has partial continuous derivative and are derivable, then derivative exists, and there holds:

b.  If function is differentiable, and functions , , and are also differentiable, then composite function is partially derivable by and , and there holds:

.

c.  Chain Rule - general case

If functions , are differentiable in the neighborhood of , and function is differentiable in the neighborhood of when , then the function composition

is differentiable at , and