Suppose that z = f(x,y), where x and y themselves depend on one or more variables. Multivariable Chain Rules allow us to differentiate z with respect to any of the variables involved:
Let x = x(t) and y = y(t) be differentiable at t and suppose that z = f(x,y) is differentiable at the point (x(t),y(t)). Then z = f(x(t),y(t)) is differentiable at t and
Although the formal proof is not trivial, the variable-dependence diagram shown here provides a simple way to remember this Chain Rule. Simply add up the two paths starting at z and ending at t, multiplying derivatives along each path.
ExampleLet z = x2y-y2 where x and y are parametrized as x = t2 and y = 2t. Then
We now suppose that x and y are both multivariable functions.
Let x = x(u,v) and y = y(u,v) have first-order partial derivatives at the point (u,v) and suppose that z = f(x,y) is differentiable at the point (x(u,v),y(u,v)). Then f(x(u,v),y(u,v)) has first-order partial derivatives at (u,v) given by
Again, the variable-dependence diagram shown here indicates this Chain Rule by summing paths for z either to u or to v.
ExampleLet z = ex2y, where x(u,v) = √[uv] and y(u,v) = 1/v. Then
These Chain Rules generalize to functions of three or more variables in a straight forward manner.
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