What is the multivariable chain rule?

The multivariable chain rule is a formula in calculus used to find the derivative of the composition of multiple functions. In the multivariable chain rule, a function expressed as y(w(x)) can be differentiated using the equation dy/dx = (dy/dw)*(dw/dx), where d indicates the derivative of the variable.

For example, the equation y = sin(10x) can be solved using the multivariable chain rule by substituting w for 10x. After the substitution, the equation can be written as y(w(x))=sin(w(x)). Using the chain rule to solve this equation results in (dsin(w)/dw) * (dw/dx). The equation can then be further simplified to cos(w)*10 = 10cos(10x).