Chain rule in several variables

With one variable, \((f\circ g)'=f'(g)\,g'\). With several, each intermediate variable is a path by which the parameter \(t\) reaches \(z\), and the contributions add up. A tree diagram helps: branches go from \(z\) to \(x\) and to \(y\), and from each one to \(t\); multiply the derivatives along each path and add the paths. With \(z=x^2y\), \(x=t\), \(y=t^2\) one gets \(2xy+x^2\cdot2t=4t^3\) after substituting, the same as differentiating \(z=t^4\).

If the intermediate variables depend on two parameters, \(x=x(s,t)\) and \(y=y(s,t)\), there are two formulas with partials: \(z_s=f_xx_s+f_yy_s\) and \(z_t=f_xx_t+f_yy_t\). In general there is one term per intermediate variable, and the formula holds for any number of variables.

In vector form, if \(\mathbf{r}(t)=\langle x(t),y(t)\rangle\), then \(\frac{d}{dt}f(\mathbf{r}(t))=\nabla f\cdot\mathbf{r}'(t)\). This is why the directional derivative uses the dot product with the gradient, and why, if \(f\) is constant along the curve, \(\nabla f\perp\mathbf{r}'\). The same idea gives implicit differentiation: from \(F(x,y)=0\) with \(y=y(x)\) we get \(\frac{dy}{dx}=-\frac{F_x}{F_y}\) if \(F_y\ne0\).

Typical errors are dropping a term, using \(y_t\) where \(y_s\) belongs, or evaluating the partials of \(f\) at the wrong point: \(f_x\) and \(f_y\) are evaluated at \((x(t),y(t))\). The rule requires \(f\) to be differentiable; the mere existence of the partials is not enough.