Subtleties of the Chain Rule Let w=f(x, y, z)=2 x+3 y+4 z...

Subtleties of the Chain Rule Let $w=f(x, y, z)=2 x+3 y+4 z$ which is defined for all $(x, y, z)$ in $\mathbb{R}^{3}$. Suppose that we are interested in the partial derivative $w_{x}$ on a subset of $\mathbb{R}^{3}$, such as the plane $P$ given by $z=4 x-2 y .$ The point to be made is that the result is not unique unless we specify which variables are considered independent. a. We could proceed as follows. On the plane $P$, consider $x$ and $y$ as the independent variables, which means $z$ depends on $x$ and $y,$ so we write $w=f(x, y, z(x, y)) .$ Differentiate with respect to $x$ holding $y$ fixed to show that $\left(\frac{\partial w}{\partial x}\right)_{y}=18,$ where the subscript $y$ indicates that $y$ is held fixed. b. Alternatively, on the plane $P$, we could consider $x$ and $z$ as the independent variables, which means $y$ depends on $x$ and $z,$ so we write $w=f(x, y(x, z), z)$ and differentiate with respect to $x$ holding $z$ fixed. Show that $\left(\frac{\partial w}{\partial x}\right)_{z}=8,$ where the subscript $z$ indicates that $z$ is held fixed. c. Make a sketch of the plane $z=4 x-2 y$ and interpret the results of parts (a) and (b) geometrically. d. Repeat the arguments of parts (a) and (b) to find $\left(\frac{\partial w}{\partial y}\right)$ $\left(\frac{\partial w}{\partial y}\right)_{z},\left(\frac{\partial w}{\partial z}\right)_{x},$ and $\left(\frac{\partial w}{\partial z}\right)_{y}$.

Subtleties of the Chain Rule Let $w=f(x, y, z)=2 x+3 y+4 z$ which is defined for all $(x, y, z)$ in $\mathbb{R}^{3}$. Suppose that we are interested in the partial derivative $w_{x}$ on a subset of $\mathbb{R}^{3}$, such as the plane $P$ given by $z=4 x-2 y .$ The point to be made is that the result is not unique unless we specify which variables are considered independent.
a. We could proceed as follows. On the plane $P$, consider $x$ and $y$ as the independent variables, which means $z$ depends on $x$ and
$y,$ so we write $w=f(x, y, z(x, y)) .$ Differentiate with respect
to $x$ holding $y$ fixed to show that $\left(\frac{\partial w}{\partial x}\right)_{y}=18,$ where the subscript $y$ indicates that $y$ is held fixed.
b. Alternatively, on the plane $P$, we could consider $x$ and $z$ as the independent variables, which means $y$ depends on $x$ and $z,$ so we write $w=f(x, y(x, z), z)$ and differentiate with respect to
$x$ holding $z$ fixed. Show that $\left(\frac{\partial w}{\partial x}\right)_{z}=8,$ where the subscript
$z$ indicates that $z$ is held fixed.
c. Make a sketch of the plane $z=4 x-2 y$ and interpret the results of parts (a) and (b) geometrically.
d. Repeat the arguments of parts (a) and (b) to find $\left(\frac{\partial w}{\partial y}\right)$ $\left(\frac{\partial w}{\partial y}\right)_{z},\left(\frac{\partial w}{\partial z}\right)_{x},$ and $\left(\frac{\partial w}{\partial z}\right)_{y}$.

Key Concepts

Notation for Partial Derivatives with Held Variables

When computing partial derivatives in the presence of a constraint, the notation (for example, ?w/?x with a subscript indicating the variable held constant) explicitly states which variable is being treated as fixed. This is essential for clarity because different choices of independent variables can lead to different numerical results for the derivative.

Geometric Interpretation on Constraint Surfaces

This concept involves visualizing how a function behaves when restricted to a surface defined by a constraint. The different choices of independent variables correspond to taking slices through the surface in different coordinate directions, leading to different tangent slopes. This helps one understand that the geometric meaning of a derivative depends on the direction in which the function is viewed or varied.

Chain Rule in Multivariable Calculus

This concept refers to the method used for differentiating a function of several variables that depend on other variables. It requires taking into account how changes in independent variables indirectly affect the function via intermediate variables. When a variable is expressed in terms of other variables by way of a constraint, the chain rule helps compute partial derivatives correctly by accounting for these dependencies.

Dependent and Independent Variables

In problems where a constraint (such as a plane or surface) is present, not all variables are free to vary independently. One must decide which variables to treat as independent and which as functions of the others. This choice affects the computation of partial derivatives because it determines which other variables are held constant during differentiation.

Transcript

00:01 According to this question, we use chain rule and partial derivative of omega function.

00:06 We have given partially derivative omega is equal to f of x, comma, y, comma, z is equal to 2x plus 3y plus 4z.

00:25 D differentiation with respect to x and z.

00:30 When we differentiate with respect to x, there is f x, x, comma, y, z is equal to f of x.

00:43 D differentiation with respect to x, this equation plus 4 z, we know that when we differentiate with respect to x, the result is only 2.

00:56 When we differentiate with respect to z, the function f of x, x, comma, y, z, there is differentiation with respect to z.

01:09 2x plus 3y plus 4 z is equal to f of z is equal to 4.

01:19 Partially differentiate z is equal to 4 x minus 2y with respect to x curly upon curly x z is equal to curly upon curly x 4 x minus 2y the differentiation of curly z upon curly x is equal to 4.

01:52 We know that differentiation of this function with respect to x is only 4.

01:58 There is partially derivative of w is equal to f of x, x, x, y, z, x, y, differentiation with respect to x by holding holding the value of y fixed when we use dirt we differentiate with respect to y and why holding curly omega upon curly x differentiation with respect to x holding y is equal to curly upon curly x f of x comma y comma z x x comma y differentiation with respect to x holding the value of y is fixed the curly omega upon curly x is holding of y is equal to curly upon curly x f of x comma y comma z x comma y there is a function we have the value f of x plus f of z x comma y there is a function we have the value f of x plus f of z curly z upon curly x.

03:22 We easy to solve.

03:24 We substitute the values.

03:27 Substitute to 4 f of x and 4 4 f of z and 4 4 curly x and 4 curly x in the equation.

03:50 When we put, we have curly omega upon curly x...

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