Math1231 Partial Derivative Chain Rules

Three Variable Functions

(1)
\begin{align} \begin {array} {r l} z & = F(x,y) \\ x & = x(t) \\ y & = y(t) \\ \frac {dz}{dt} & = \frac {\partial F}{\partial x} \frac{dx}{dt} + \frac {\partial F}{\partial y} \frac {dy}{dt} \end {array} \end{align}

Four Variable Functions

(2)
\begin{align} \begin {array} {r l} Z &= F(x,y) \\ x & = x(s,t) \\ y & = y(s, t) \\ \frac {\partial F}{\partial t} & = \frac{\partial F}{\partial x} \frac {\partial x}{\partial t} + \frac {\partial F}{\partial y} \frac{\partial y}{\partial t} \\ \frac {\partial F}{\partial s} & = \frac {\partial F}{\partial x} \frac {\partial x}{\partial s} + \frac{\partial F}{\partial y} \frac {\partial y}{\partial s} \\ \end {array} \end{align}

Differences

The Three Variable Function is a derivative, not a partial derivative, whereas the Four Variable Function is the opposite, due to the fact that you’re not differentiating for all the variables, hence by definition.

Example

(3)
\begin{align} \begin {array} {r l} F(x, y) & = x^2y \\ x(s, t) & = se^t \\ y(s, t) & = s + t \\ \end {array} \end{align}
(4)
\begin{align} \begin {array} {l || l} \frac {\partial F}{\partial s}: & \frac {\partial F}{\partial t}: \\ = (2xy * e^t) + (x^2 * 1) & = 2xy * ste^t + x^2 \\ = (2se^{t}*(s+t)*e^t) + (s^2e^{2t}) &= (2se^t * (s+t) * ste^t) + s^2e^{2t} \\ = (e^{2t}(2s(s+t) + s^2) & = (e^{2t}*(2s(s+t)*st)) + s^2 \\ = (e^{2t}(2s^2 + 2st + s^2) & = (e^{2t}((2s^2 + 2st)st + s^2) \\ = (e^{2t}s(2s + 2t + s) & = ( e^{2t}(2s^{3}t + 2s^{2}t^{2} + s^2) \\ = (e^{2t}s(3s + 2t) & = (e^{2t}s^{2}(2st + 2t^2 + 1) \\ \end {array} \end{align}