Partial Derivatives of Multivariable Functions

1. The Concept of a Partial Derivative

With a function like $f(x)$, the derivative $f^{\prime }(x)$ tells us how the function's output changes as its single input $x$ changes. But what about a function with two inputs, like $f(x,y)$? How does its output change? The answer depends on whether we are changing $x$, changing $y$, or changing both.

The partial derivative is our tool for handling the simplest case: changing only one variable at a time while holding all others constant.

To build your intuition, let's start with a video that introduces this concept and the new notation you'll be using.

As the video explained, the central idea is to simplify the problem. By holding $y$ constant, the multivariable function $f(x,y)$ temporarily behaves like a single-variable function of $x$.

We can visualize this beautifully. Imagine a function $f(x,y)$ as a 3D surface. Taking a partial derivative with respect to $x$ is like taking a vertical slice of that surface parallel to the x-axis. Along that slice, $y$ is constant. The partial derivative $\frac{\partial f}{\partial x}$ is then simply the slope of the curve on that slice.

Notation and Definition

Formally, the partial derivative of $f$ with respect to $x$ is defined using a limit, just like in single-variable calculus. Notice how only $x$ is changed by $h$.

There are several common notations for partial derivatives of a function $z=f(x,y)$. They all mean the same thing.

• Partial derivative with respect to x: $\frac{\partial f}{\partial x}$, $f_x$, $\frac{\partial z}{\partial x}$, $D_{x}f$
• Partial derivative with respect to y: $\frac{\partial f}{\partial y}$, $f_y$, $\frac{\partial z}{\partial y}$, $D_{y}f$

For a more detailed look at the notation and formal definition, you can refer to Paul's Online Math Notes.

2. The Mechanics of Computation

You now understand the theory: to find a partial derivative, you hold all other variables constant. The practical consequence of this is straightforward:

To compute $\frac{\partial f}{\partial x}$, treat every occurrence of $y$ (and any other variables) as if it were a number, like 5 or $\pi$. Then, apply all the differentiation rules you already know.

Let's see this in action. This next video is a fantastic resource that walks through numerous examples, from basic polynomials to more complex functions involving exponentials, logs, and trigonometric functions.

A Quick Example to Reinforce

Let's do one together. Consider the function $f(x,y)=x^3{e}^{y^2}$.

1. Find $\frac{\partial f}{\partial x}$:

   • We are differentiating with respect to $x$.
   • This means we treat $y$ as a constant. Therefore, the entire term $e^{y^2}$ is just a constant multiplier.
   • The problem is like finding the derivative of $C\cdot x^3$, where $C=e^{y^2}$.
   • The derivative of $x^3$ is $3x^2$.
   • So, $\frac{\partial f}{\partial x}=3x^2\cdot e^{y^2}$.

2. Find $\frac{\partial f}{\partial y}$:

   • We are differentiating with respect to $y$.
   • This means we treat $x$ as a constant. Therefore, the term $x^3$ is a constant multiplier.
   • We need the derivative of $e^{y^2}$ with respect to $y$. This requires the chain rule!
   • Derivative of outer function ($e^u$): $e^u$.
   • Derivative of inner function ($y^2$): $2y$.
   • So, the derivative of $e^{y^2}$ is $e^{y^2}\cdot 2y$.
   • Putting it all together: $\frac{\partial f}{\partial y}=x^3\cdot (e^{y^2}\cdot 2y)=2y{x}^3{e}^{y^2}$.

Notice how all the rules from our previous lesson—the power rule, the rule for exponentials, and the chain rule—were used directly.

3. Combining Rules and Extending to More Variables

The true power of this method becomes apparent with more complex functions, where you might need the product or quotient rule. A key skill is identifying when these rules are necessary.

For a function like $f(x,y)=x^2{y}^3$:

• When finding $\frac{\partial f}{\partial x}$, $y^3$ is a constant. No product rule needed.
• When finding $\frac{\partial f}{\partial y}$, $x^2$ is a constant. No product rule needed.

But for a function like $g(x,y)=x^2{e}^{xy}$:

• When finding $\frac{\partial g}{\partial x}$, both $x^2$ and $e^{xy}$ contain $x$. You must use the product rule.

The same logic applies to functions of three or more variables, like $f(x,y,z)$. To find $\frac{\partial f}{\partial y}$, you simply treat both $x$ and $z$ as constants.

Conclusion

In this lesson, we have extended our knowledge of differentiation into the realm of multivariable functions. You've learned that the seemingly complex task of finding a partial derivative simplifies to applying the rules you already know, with the crucial discipline of treating all other variables as constants.

Key Takeaways:

• A partial derivative measures the rate of change of a multivariable function with respect to one variable, while all other variables are held constant.
• The notation $\frac{\partial f}{\partial x}$ or $f_x$ denotes the partial derivative of $f$ with respect to $x$.
• The core computational technique is to treat all variables—except the one you are differentiating with respect to—as constants.
• All your existing rules (Power, Product, Quotient, Chain) apply directly.

Preview of the Next Lesson:
We've learned how to find the rate of change in the x-direction ($\frac{\partial f}{\partial x}$) and the y-direction ($\frac{\partial f}{\partial y}$). But what if we want to know the direction of the steepest possible uphill slope on our 3D surface? This is a vital question for optimization. In our next lesson, "Explain the concept of a gradient vector and its relation to the direction of steepest ascent," you will learn how to combine all the partial derivatives of a function into a single object called the gradient vector. This vector points in the direction of the fastest increase of the function and is the absolute cornerstone of how we train almost all modern AI models.