What Is Partial Derivative? Definition, Rules and Examples

01.20.2022 • 4 min read

Rachel McLean

Subject Matter Expert

This article is an overview of partial derivatives. Learn the definition of partial derivatives, how to do partial differentiation, and practice with some examples.

We use partial differentiation to differentiate a function of two or more variables. For example,

$f(x, y) = xy + x^2y$

is a function of two variables.

If we want to find the partial derivative of a two-variable function with respect to $x$, we treat $y$ as a constant and use the notation $\frac{\partial{f}}{\partial{x}}$.
If we want to find the partial derivative of a two-variable function with respect to $y$, we treat $x$ as a constant and use the notation $\frac{\partial{f}}{\partial{y}}$.

You can think of $\partial$ as the partial derivative symbol, sometimes called “del.” When you see this symbol, it shows that we’re taking a partial derivative.

This notation should look familiar — it’s just like the derivative of a function in Leibniz’s notation, expressed $\frac{dy}{dx}$, except that the letter “d” is replaced by $\partial$, a stylized curly $d$.

In calculus, derivatives measure the rate of change of a function with respect to a change in its input variable $x$. Since the input of a multivariable function is more than one variable, we call $\frac{\partial{f}}{\partial{x}}$ and $\frac{\partial{f}}{\partial{y}}$ partial derivatives because they only reveal the rate of change of $f$ when one variable changes, instead of both.

The partial derivative allows us to understand the behavior of a multivariable function when we let just one of its variables change, while the rest stay constant.

How to Do Partial Derivatives

How do partial derivatives work? To find a partial derivative, we find the derivative of a function of two or more variables by treating one of the variables as a constant.

To find $\frac{\partial{f}}{\partial{x}}$:

Treat $y$ as a constant.

Differentiate the function normally.

To find $\frac{\partial{f}}{\partial{y}}$:

Treat $x$ as a constant.

Differentiate the function normally.

After we designate one variable as a constant, we can use the derivative rules that are already familiar to us to differentiate the function.

Partial Derivative Rules

Derivative rules help us differentiate more complicated functions by breaking them into pieces. Here are some of the most common derivative rules to know:

The quotient rule: $\frac{d}{dx}\frac{f(x)}{g(x)} = \frac{g(x)f’(x)-f(x)g’(x)}{(g(x))^2}$

Sum/Difference Rule

The sum/difference rule: $\frac{d}{dx}(f(x) \pm g(x)) = f’(x) \pm g’(x)$

Trigonometry Rules

$\frac{d}{dx}(\sin{(x)}) = \cos{(x)}$

$\frac{d}{dx}(\cos{(x)}) = -\sin{(x)}$

$\frac{d}{dx}(\tan{(x)}) = \sec ^2 (x)$

Logarithmic and Exponential Rules

$\frac{d}{dx} (\ln{x}) = \frac{1}{x}$

$\frac{d}{dx}(e^x) = e^x$

You can view more about these rules in an explanation by one of our instructors Dr. Tim Chartier.

For example, let’s take another look at the function $f(x, y) = xy + x^2y$. Suppose we want to find $\frac{\partial{f}}{\partial{x}}$, the partial derivative with respect to $x$. The first thing to do is treat $y$ as a constant.

What does it mean to treat $y$ as a constant? A constant is a fixed, unchanging value. Examples of constants are 1, 3.5, 17, and 100,000. To treat $y$ as a constant, we imagine that $y$ is any non-zero constant value you choose.

We can do this because of the constant rule, which states that the derivative of any constant is 0.

To make it easier to imagine $y$ as a constant, we can replace $y$ with $c$ or $k$, which are two variables that are commonly used to represent constant values.
Using this trick and replacing $y$ with $k$, we have

$f(x, y) = kx + kx^2$

Now, we can find the partial derivative $\frac{\partial{f}}{\partial{x}}$ using the derivative rules. Remember to change $k$ back to $y$ when you have your final answer.

$\frac{\partial{(kx+kx^2)}}{\partial x}=k+2kx$

$\frac{\partial{(xy + x^2y)}}{\partial{x}} = y + 2yx$

We can do the same to find $\frac{\partial{f}}{\partial{y}}$, the partial derivative with respect to $y$. The first thing to do is treat $x$ as a constant. Remember that the square of any constant is simply another constant.

$f(x, y) = ky + k^2y$

Now, we can find the partial derivative $\frac{\partial{f}}{\partial{y}}$ using the derivative rules. Remember to change $k$ back to $x$ when you have your final answer.

$\frac{\partial{(ky + k^2y)}}{\partial{y}} = k + k^2$

$\frac{\partial{(xy + x^2y)}}{\partial{y}} = x + x^2$

Partial Derivatives Examples

Let’s take a look at some more partial derivative examples.

Example 1

Find the partial derivatives of $f(r, h) = \pi r^2h$.

Solution:

This function represents the volume of a cylinder. When we find the partial derivative $\frac{\partial{(\pi r^2h)}}{\partial{r}}$, we find the rate of change of the cylinder’s volume as only the radius changes.

When we find the partial derivative $\frac{\partial{(\pi r^2h)}}{\partial{h}}$, we find the rate of change of the cylinder’s volume as only the height changes. So,

Find the partial derivatives of $f(x, y, z) = xy^3 - zx + z$.

Solution:

How do partial derivatives work in more than two variables? Just the same! For a function with three variables, we change only one variable and treat the other two as constants. So,

Find the partial derivatives of $f(x, y) = x^2\sin{(y)} - y^2\cos{(x)}$.

Solution:

We can use the trigonometry derivative rules for this problem. Remember that $\sin{(y)}$ acts as a constant when we calculate $\frac{\partial{(f(x, y))}}{\partial{x}}$, and $\cos{(x)}$ acts as a constant when we calculate $\frac{\partial{(f(x, y))}}{\partial{y}}$. So,

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