Calculus IV: The Chain Rule

February 2007
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Feb 14

Calculus IV, Classes Add comments

In class on Tuesday, we stated and used the chain rule for partial derivative:

For $z=z(x_1, x_2, \ldots, x_n)$ be a function of $n$ variables, such that $x_i = x_i(t_1, t_2, \ldots, t_m)$ is a function of $m$ variables for each $i = 1, \ldots, n$ then

$\displaystyle \frac{\partial z}{\partial t_j} = \sum_{i=1}^n \frac{\partial z}{\partial x_i} \frac{\partial x_i}{\partial t_j}$ for each $j=1,\ldots, m$

We derived the result called the Implicit Differentiation theorem which gives you a shortcut to perform (single-variable) implicit differentiation using partial derivatives.

We just got started talking about directional derivatives. Next time will define the gradient and begin to show some simple useful results involving it. Then, we’ll be able to begin studying optimization of functions of several variables.

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