chain rule

\(f'(u(x)) = f'(u)u'(x)\) – "The car's position \(f\) can be given as a function of the bicycle \(u\). The car goes 10 times as fast as the bicycle. The bicycle is \(u'(x)\) fast."

1. multivariate chain rule:

For independent variables \(t=(t_1,...,t_n)\), and \(m\) functions \(x_i\), where each \(x_i = f_i(t_1,...,t_n)\), let \(y\) be a function of \(x\): \(f(x_1,...,x_n)\). Then: \[ \frac{\partial f}{\partial t_j} = \sum_{i=1}^{m} \frac{\partial x_i}{\partial t_j} \]

See this answer which has discussion of differentials and this textbook