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The Chain Rule For Several Variables

April 10th, 2009 |

Author: Merrin

Example 1. Given a multi-variable function, f(x,y), of which the variables x and y are themselves functions of t.

$f(x(t),y(t))$

then what is the derivative of f(x,y) with respect to t by the chain rule?

Solution 1. There are parts corresponding to each variable in a sum.

$\cfrac{\partial f(x,y)}{\partial t} =\cfrac{\partial f}{\partial x} \cfrac{\partial x}{\partial t} +\cfrac{\partial f}{\partial y} \cfrac{\partial y}{\partial t}$

Also as an added bonus, f(x,y) might also have some explicit time dependence in which case adding another piece to the mix is necessary.

$\cfrac{df(x(t),y(t),t)}{dt} =\cfrac{\partial f}{\partial t} +\cfrac{\partial f}{\partial x} \cfrac{\partial x}{\partial t} +\cfrac{\partial f}{\partial y} \cfrac{\partial y}{\partial t}$

Note that I have written the derivative as our normal

$\cfrac{d}{dt}$

operator for single variables. This is because what we have taken is a total derivative.

Sometimes it is explicitly known that x and y are only functions of t in which case we could write regular derivative on the terms of the end of the chain rule.

$\cfrac{df(x,y,t)}{dt} =\cfrac{\partial f}{\partial t} +\cfrac{\partial f}{\partial x} \cfrac{dx}{dt} +\cfrac{\partial f}{\partial y} \cfrac{dy}{dt}$

Example 2. How about three variables?

Solution 2. For three variables there is a logical generalization,

$\cfrac{d}{dt} f(x(t),y(t),z(t),t)=\cfrac{\partial f}{\partial t} +\cfrac{\partial f}{\partial x} \cfrac{dx}{dt} +\cfrac{\partial f}{\partial y} \cfrac{dy}{dt} +\cfrac{\partial f}{\partial z} \cfrac{dz}{dt}$

Example 3. Given a function below which is the probably the most complicated anyone will ever encounter, take the total derivative.

$f\left(x\left(u(t),v(t)\right),y\left(r(t),s(t)\right)\right)$

Solution 3. By summing up piece by piece and using the chain rule for each layer of variables the answer emerges.

${\cfrac{df}{dt} } = {\cfrac{\partial f}{\partial x} \cfrac{\partial x}{\partial t} +\cfrac{\partial f}{\partial y} \cfrac{\partial y}{\partial t} } \\ f'(t)= {\cfrac{\partial f}{\partial x} \left(\cfrac{\partial x}{\partial u} \cfrac{\partial u}{\partial t} +\cfrac{\partial x}{\partial v} \cfrac{\partial v}{\partial t} \right)+\cfrac{\partial f}{\partial y} \left(\cfrac{\partial y}{\partial r} \cfrac{\partial r}{\partial t} +\cfrac{\partial y}{\partial s} \cfrac{\partial s}{\partial t} \right)}$