Partial Derivative Calculator
Calculate $\frac{\partial f}{\partial x}$, $\frac{\partial^2 f}{\partial x \partial y}$, and evaluate at points.
$$ \frac{\partial}{\partial x} (x^2 + y^2) $$
Function f(x, y, …)
Respect to (e.g. x, y)
At Point (Optional)
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Result
3D Surface Visualization (z = f(x,y))
Detailed Step-by-Step Solution
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By Prof. David Anderson
Professor of Mathematics | 20+ Years Experience
“In Single-Variable Calculus, finding a derivative is like walking along a line. In Multivariable Calculus, it’s like standing on a 3D hill. Which way is down? North? East? That’s where the Partial Derivative comes in. It measures how a function changes as you move in just one direction while keeping everything else frozen. I designed this Partial Derivative Calculator to help you visualize that ‘frozen’ state and solve complex gradients instantly.”
The Ultimate Guide to Partial Derivatives: First Order, Mixed, and Implicit Differentiation
How to Use a Partial Derivative Calculator for Multivariable Functions
A Partial Derivative is the derivative of a function of multiple variables ($x, y, z…$) with respect to one variable, while treating the others as constants. This concept is the bedrock of Multivariable Calculus, vector calculus, physics, and optimization.
Whether you are calculating the slope of a surface in the x-direction, finding the Gradient Vector, or solving Second Order Mixed Derivatives ($f_{xy}$), our Partial Derivative Calculator handles the symbolic differentiation step-by-step using the standard “Hold Constant” rule.
1. Understanding the Notation ($\partial$)
Unlike ordinary derivatives ($d$), Partial Differentiation uses the symbol $\partial$ (pronounced “del” or “curly d”).
Definition Limit
$$ \frac{\partial f}{\partial x} = \lim_{h \to 0} \frac{f(x+h, y) – f(x, y)}{h} $$
Measures change in $x$, holding $y$ constant.
2. The Golden Rule: “Hold It Constant”
The secret to manual calculation (and how our Partial Derivative Calculator works) is simple: If you are differentiating with respect to $x$, pretend $y$ is a number like 5.
Step 1
Identify the Target Variable
Look at the notation: $\frac{\partial f}{\partial x}$. The bottom variable ($x$) is your target for Partial Differentiation. All other letters ($y, z, t$) are effectively constants.
Step 2
Apply Derivative Rules
Use standard Power, Product, and Chain rules on the target variable. Treat other variables as coefficients.
Example: $\frac{\partial}{\partial x}(y x^2) = y \cdot 2x = 2xy$
Step 3
Clean Up
Simplify the expression. Note that if a term contains ONLY $y$ (no $x$), its derivative with respect to $x$ is Zero.
3. Master Class: First Order Examples
Type A: Polynomial Function
Beginner
Calculate $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ for $f(x, y) = x^3 + x^2y^3 – 2y^2$ using our Partial Derivative Solver.
1. Differentiate w.r.t x (Treat y as constant)
$$ f_x = \frac{\partial}{\partial x}(x^3) + \frac{\partial}{\partial x}(x^2 \cdot y^3) – \frac{\partial}{\partial x}(2y^2) $$
$$ = 3x^2 + (2x) \cdot y^3 – 0 $$
$$ = 3x^2 + 2xy^3 $$
2. Differentiate w.r.t y (Treat x as constant)
$$ f_y = 0 + x^2 \cdot (3y^2) – 4y $$
$$ = 3x^2y^2 – 4y $$
4. Higher Order & Mixed Partial Derivatives
You can differentiate a function twice. The interesting case is the Mixed Partial Derivative $f_{xy}$ (differentiate by $x$, then by $y$).
Concept
Clairaut’s Theorem
For most continuous functions, the order of differentiation does not matter. The Mixed Derivatives are equal:
$$ f_{xy} = f_{yx} $$
$$ \frac{\partial^2 f}{\partial y \partial x} = \frac{\partial^2 f}{\partial x \partial y} $$
Example
Calculating $f_{xy}$
If $f = x^2 y^3$:
1. $f_x = 2xy^3$
2. Take $\frac{\partial}{\partial y}$ of $(2xy^3)$
3. Result: $2x(3y^2) = 6xy^2$
1. $f_x = 2xy^3$
2. Take $\frac{\partial}{\partial y}$ of $(2xy^3)$
3. Result: $2x(3y^2) = 6xy^2$
5. Advanced: The Gradient & Implicit Differentiation
Vector
The Gradient ($\nabla f$)
The Gradient Vector contains all first-order partial derivatives. It points in the direction of steepest ascent.
$$ \nabla f = \langle \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \rangle $$
Rule
Implicit Differentiation
To find $dy/dx$ for an equation $F(x,y)=0$ without isolating $y$, use the formula involving partials:
$$ \frac{dy}{dx} = -\frac{F_x}{F_y} $$
6. Comparison: Types of Derivatives
| Type | Symbol | Used For… | Key Concept |
|---|---|---|---|
| Ordinary | $d/dx$ | Single variable functions | Slope of tangent line |
| Partial | $\partial/\partial x$ | Multivariable functions | Slope in one direction |
| Gradient | $\nabla f$ | Vector fields / Optimization | Direction of max change |
| Total | $dz$ | Error analysis | Approximate total change |
7. Professor’s FAQ
How do I use the Chain Rule with partial derivatives?
If $z = f(x, y)$ and $x(t), y(t)$ are functions of time, the Multivariable Chain Rule is:
$\frac{dz}{dt} = \frac{\partial f}{\partial x}\frac{dx}{dt} + \frac{\partial f}{\partial y}\frac{dy}{dt}$. This is essential in physics.
Does order matter for mixed derivatives?
According to Clairaut’s Theorem, if the functions are continuous, the order does NOT matter ($f_{xy} = f_{yx}$). If you calculate them and get different answers, check your algebra—or the function is discontinuous at that point.
What is the notation $f_{xx}$?
This is subscript notation for a Second Order Partial Derivative. It means “differentiate with respect to x, then differentiate that result with respect to x again.” It corresponds to $\frac{\partial^2 f}{\partial x^2}$.
References & Further Reading
- Stewart, J. (2020). Calculus: Early Transcendentals (9th ed.). Cengage Learning. (Chapter 14: Partial Derivatives).
- Paul’s Online Math Notes. “Partial Derivatives.” Lamar University.
- MIT OpenCourseWare. “Multivariable Calculus.” 18.02.
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