First-Order Partial Derivatives
students, imagine you are looking at a landscape shaped by more than one input 🌄. The height of a hill might depend on where you stand east-west and north-south, so instead of one variable, you have two or more. In multivariable calculus, this is where partial derivatives begin. The first step is learning how to measure change in just one direction at a time.
What are first-order partial derivatives?
A first-order partial derivative measures how a multivariable function changes when one variable changes and the other variables are held constant. If a function is $f(x,y)$, then:
- the partial derivative with respect to $x$ is written as $\frac{\partial f}{\partial x}$ or $f_x$
- the partial derivative with respect to $y$ is written as $\frac{\partial f}{\partial y}$ or $f_y$
The word partial matters because we are not looking at the total change from every input at once. Instead, we focus on one input while treating the others as constants.
For example, if $f(x,y)=x^2y+3y$, then:
$$\frac{\partial f}{\partial x}=2xy$$
because $y$ is treated as a constant, and
$$\frac{\partial f}{\partial y}=x^2+3$$
because $x$ is treated as a constant.
This is similar to asking: “If I move only east, how does the elevation change?” while ignoring north-south movement for the moment. 🚶♂️
How to compute first-order partial derivatives
To compute a first-order partial derivative, use ordinary differentiation rules, but hold all other variables constant. This means rules like the power rule, product rule, and chain rule still work, as long as you remember which variables are fixed.
Example 1: Polynomial function
Let $f(x,y)=4x^3y^2-7xy+5$.
To find $f_x$, treat $y$ as a constant:
$$f_x(x,y)=12x^2y^2-7y$$
To find $f_y$, treat $x$ as a constant:
$$f_y(x,y)=8x^3y-7x$$
Notice that the constant $5$ disappears in both cases because the derivative of a constant is $0$.
Example 2: Trigonometric function
Let $g(x,y)=\sin(xy)$.
To find $g_x$, use the chain rule. Since the inside function is $xy$ and $y$ is constant with respect to $x$:
$$g_x(x,y)=y\cos(xy)$$
To find $g_y$, now $x$ is constant:
$$g_y(x,y)=x\cos(xy)$$
This example shows an important idea: a first-order partial derivative can involve both the variable being differentiated and the variables held constant.
Interpreting first-order partial derivatives
First-order partial derivatives are not just calculation rules. They describe rates of change.
If $f(x,y)$ represents temperature, then:
- $f_x(x,y)$ tells how fast temperature changes as $x$ changes
- $f_y(x,y)$ tells how fast temperature changes as $y$ changes
If $f(x,y)$ represents profit for a business, where $x$ might be number of ads and $y$ might be price, then:
- $f_x$ tells how profit changes when the number of ads changes, keeping price fixed
- $f_y$ tells how profit changes when price changes, keeping ad count fixed
In real life, this is useful because it isolates one factor at a time. Businesses, engineers, and scientists often need to know which input has the strongest effect when others are held steady 📈.
A first-order partial derivative can be positive, negative, or zero:
- if $f_x>0$, then increasing $x$ increases $f$
- if $f_x<0$, then increasing $x$ decreases $f$
- if $f_x=0$, then changing $x$ locally has no immediate effect on $f$
The same idea applies to $f_y$ and other variables.
The notation and what it means
There are several standard notations for first-order partial derivatives:
$$\frac{\partial f}{\partial x}, \quad f_x, \quad \partial_x f$$
All of these mean the same thing: the derivative of $f$ with respect to $x$, while other variables stay fixed.
If the function has more variables, such as $f(x,y,z)$, then first-order partial derivatives include:
$$f_x, \quad f_y, \quad f_z$$
For example, if $f(x,y,z)=x^2yz+e^z$, then:
$$f_x=2xyz$$
$$f_y=x^2z$$
$$f_z=x^2y+e^z$$
Each derivative focuses on one variable at a time.
A closer look at the “hold other variables constant” idea
This idea is the heart of first-order partial derivatives. Suppose a function describes the surface of a trampoline. If you want to know how the surface changes as you move left-right, you freeze the forward-backward position. That creates a one-variable slice through the surface.
For $f(x,y)$, when you compute $f_x$, you are imagining $y$ fixed at some number, say $y=2$. Then the function becomes a one-variable function of $x$ only.
For instance, if $f(x,y)=x^2+xy$, and you fix $y=2$, then:
$$f(x,2)=x^2+2x$$
Now differentiate with respect to $x$:
$$\frac{d}{dx}f(x,2)=2x+2$$
This matches the partial derivative:
$$f_x(x,y)=2x+y$$
and when $y=2$, it becomes $2x+2$.
This “slice” viewpoint is one of the best ways to understand partial derivatives. You are studying the function along one direction while keeping the others constant.
First-order partials and graphs of surfaces
For a function $f(x,y)$, the graph is usually a surface in three-dimensional space. On that surface, the partial derivatives tell you about slopes in coordinate directions.
- $f_x$ gives the slope in the $x$-direction
- $f_y$ gives the slope in the $y$-direction
If both are zero at a point, the surface may be flat there in the coordinate directions, though that does not always mean the point is a maximum or minimum.
Example: Let $f(x,y)=x^2+y^2$.
Then:
$$f_x=2x$$
$$f_y=2y$$
At the point $(0,0)$:
$$f_x(0,0)=0, \quad f_y(0,0)=0$$
So the surface is flat in both coordinate directions there. This point is actually the bottom of a bowl-shaped surface. 🥣
In contrast, for $f(x,y)=x^2-y^2$:
$$f_x=2x, \quad f_y=-2y$$
At $(0,0)$, both are also zero, but the surface is saddle-shaped, not a minimum or maximum. So first-order partials give important local information, but they do not always tell the whole story.
Connection to the rest of partial derivatives
First-order partial derivatives are the foundation for the rest of the topic. Once students understands first-order partials, the next steps in multivariable calculus become easier:
- higher-order partial derivatives come from differentiating partial derivatives again, such as $f_{xx}$, $f_{xy}$, and $f_{yy}$
- differentiability and local linearity use first-order partials to build a linear approximation near a point
For example, if $f(x,y)$ is differentiable near $(a,b)$, then the first-order partial derivatives help form the tangent plane approximation:
$$f(x,y)\approx f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)$$
This formula shows why first-order partials matter so much. They are the slopes that build the best linear estimate near a point.
So first-order partial derivatives are not just a separate skill. They are the starting point for understanding shape, change, and approximation in several variables.
Conclusion
First-order partial derivatives measure how a multivariable function changes when one variable changes and the others stay fixed. They use familiar derivative rules, but with a new mindset: focus on one direction at a time. This makes them useful for interpreting slopes, rates of change, and local behavior of surfaces. They also connect directly to higher-order partials and to linear approximation later in the course. If students can compute and interpret $f_x$, $f_y$, and related derivatives, then the foundation for the rest of partial derivatives is in place ✅.
Study Notes
- A first-order partial derivative measures change with respect to one variable while other variables are held constant.
- Common notations include $\frac{\partial f}{\partial x}$, $f_x$, and $\partial_x f$.
- To compute a partial derivative, use ordinary differentiation rules and treat other variables as constants.
- Example: if $f(x,y)=x^2y+3y$, then $f_x=2xy$ and $f_y=x^2+3$.
- First-order partials are rates of change in coordinate directions.
- Positive, negative, or zero values tell whether the function increases, decreases, or is locally unchanged in that direction.
- For $f(x,y)$, the graph is a surface, and $f_x$ and $f_y$ describe slopes along the $x$- and $y$-directions.
- First-order partial derivatives are the basis for higher-order partials and for local linearity.
- A tangent plane approximation uses first-order partials: $$f(x,y)\approx f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)$$
- Understanding first-order partials helps with applications in science, engineering, economics, and data modeling.