Units

# Unit 1: Partial Derivatives

## The Concept

When studying derivatives of functions of one variable $y=f(x)$, we found that one interpretation of the derivative is an instantaneous rate of change of $y$ as a function of $x$. Leibniz notation for the derivative is $\frac{dy}{dx}$, which implies that $y$ is the dependent variable and $x$ is the independent variable. $\frac{dy}{dx}$ also represents the slope of the tangent line at a certain point of this function.

For a function $z=f(x,y)$ of two variables, $x$ and $y$ are the independent variables (input to function $f$) and $z$ is the dependent variable (output of function $z$, the value of $z$ is depend on the values of $x$ and $y$ ). We will have two partial derivatives and their Leibniz notations are $\frac{\partial z}{\partial x}$, and $\frac{\partial z}{\partial y}$. They are analogous to ordinary derivatives:

$\frac{\partial z}{\partial x}(x_0, y_0)=\frac{\text{change in } z}{\text{change in }x}(\text{holding }y \text{ as a constant } y_0)$

$\frac{\partial z}{\partial y}(x_0, y_0)=\frac{\text{change in } z}{\text{change in }y}(\text{holding }x \text{ as a constant } x_0)$

Besides the Leibniz notations above, you can also write the derivatives as $\frac{\partial z}{\partial x}=f_x$ and $\frac{\partial z}{\partial y}=f_y$. Similar to the geometric meaning of $\frac{dy}{dx}$ in two-dimensional (2D), $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$ in three-dimensional (3D) represent the slopes of tangent lines as well.

## The Plot

Now, you should engage with the 3D plot below for partial derivatives[1]. Follow the steps below to apply changes to the plot and observe the effects:

1. Input a function of two variables, then set $y$ as a constant, e.g., $-1$. A cross-section plane $y=-1$ is plotted. Recall that the function $y = y_0$ (or $x=x_0$) in 3D represents the planes that are perpendicular to the $xy$-plane.
2. A tangent line passing through the point ($x_0$, $y_0$) and also on the cross-section plane $y=y_0$ is plotted. Change the $y$-values using the slider, and you will see the cross-section and the tangent line changes. You can also rotate the graph to get a better view. Since the particle derivative is the slope of the tangent line, the partial derivative $\frac{\partial z}{\partial x}$ changes as well.
3. Repeat the same steps in (1) and (2) for $\frac{\partial z}{\partial y}$.

## Self-Checking Questions

Check your understanding by solving the following questions[2]:

1. Let $f(x,y)=\frac{xy}{x-y}$. Find $f_x(2,-2)$ and $f_y(2,-2)$.
2. The apparent temperature index, $A$, is a measure of how the temperature feels,

$A=0.885x -22.4 y +1.2 xy -0.544$

where $x$ is relative humidity and $y$ is the air temperature. Find $\frac{\partial A}{\partial x}$ and $\frac{\partial A}{\partial y}$ when $x=20°F$ and $y=1$.