Partial Derivative

Partial Derivative

A derivative of a function that has more than one independent variable. Partial derivatives are found by treating one independent variable as a variable and the rest as constants.

Notation for partial derivatives of z=f(x,y): f_x and f_y denote ∂f/∂x and ∂f/∂y. Example: f_x=3x²+7y, f_y=7x−10y.

Key Formula

\frac{\partial f}{\partial x} = \lim_{h \to 0} \frac{f(x+h,\, y) - f(x,\, y)}{h}

Where:

$f$ = A function of two or more variables
$x$ = The variable you are differentiating with respect to
$y$ = Another variable, treated as a constant during differentiation
$\partial$ = The 'curly d' symbol, used instead of 'd' to indicate a partial (not ordinary) derivative
$h$ = A small increment approaching zero

Worked Example

Problem: Find both partial derivatives of f(x, y) = 3x²y + 5xy³ − 2y.

Step 1: To find ∂f/∂x, treat y as a constant and differentiate with respect to x.

\frac{\partial f}{\partial x} = \frac{\partial}{\partial x}\bigl(3x^2 y + 5x y^3 - 2y\bigr)

Step 2: Differentiate each term: 3y · 2x = 6xy, then 5y³ · 1 = 5y³, and −2y is a constant so its derivative is 0.

\frac{\partial f}{\partial x} = 6xy + 5y^3

Step 3: To find ∂f/∂y, treat x as a constant and differentiate with respect to y.

\frac{\partial f}{\partial y} = \frac{\partial}{\partial y}\bigl(3x^2 y + 5x y^3 - 2y\bigr)

Step 4: Differentiate each term: 3x² · 1 = 3x², then 5x · 3y² = 15xy², and −2.

\frac{\partial f}{\partial y} = 3x^2 + 15xy^2 - 2

Answer: ∂f/∂x = 6xy + 5y³ and ∂f/∂y = 3x² + 15xy² − 2.

Another Example

This example involves transcendental functions (sin, exp) and the chain rule, then evaluates the partial derivative at a specific point—skills beyond simple polynomial differentiation.

Problem: Given g(x, y) = x² sin(y) + eˣʸ, find ∂g/∂x and evaluate it at the point (1, 0).

Step 1: Treat y as a constant and differentiate with respect to x. The first term x² sin(y) differentiates to 2x sin(y).

\frac{\partial}{\partial x}\bigl(x^2 \sin y\bigr) = 2x\sin y

Step 2: For the second term eˣʸ, use the chain rule. Since y is constant, the derivative of xy with respect to x is y.

\frac{\partial}{\partial x}\bigl(e^{xy}\bigr) = y\, e^{xy}

Step 3: Combine the results to get the full partial derivative.

\frac{\partial g}{\partial x} = 2x\sin y + y\,e^{xy}

Step 4: Evaluate at (1, 0): substitute x = 1 and y = 0. Note sin(0) = 0 and e⁰ = 1.

\frac{\partial g}{\partial x}\bigg|_{(1,0)} = 2(1)\sin(0) + 0 \cdot e^{0} = 0 + 0 = 0

Answer: ∂g/∂x = 2x sin(y) + y eˣʸ, which equals 0 at the point (1, 0).

Frequently Asked Questions

What is the difference between a partial derivative and an ordinary derivative?

An ordinary derivative applies to a function of a single variable and measures its total rate of change. A partial derivative applies to a function of two or more variables and measures the rate of change with respect to one variable while holding all others constant. The notation switches from d to ∂ to signal this distinction.

When do you use partial derivatives?

You use partial derivatives whenever a quantity depends on more than one variable and you want to understand how it changes with respect to just one of them. They appear throughout physics (e.g., heat equations, wave equations), economics (e.g., marginal cost with multiple inputs), machine learning (gradient descent), and engineering.

Can you take the partial derivative of a function with respect to a variable that doesn't appear in it?

Yes. If the function does not actually depend on that variable, the partial derivative with respect to it is simply zero. For example, if f(x, y) = 3x², then ∂f/∂y = 0 because x² has no y-dependence.

Partial Derivative vs. Ordinary (Total) Derivative

	Partial Derivative	Ordinary (Total) Derivative
Number of variables	Function of two or more variables	Function of one variable
Notation	∂f/∂x (curly d)	df/dx or f′(x)
What stays fixed	All variables except the one being differentiated are held constant	Nothing else to hold constant—there is only one variable
Differentiation rules	Same rules (power, product, chain, etc.), but other variables act as constants	Standard single-variable differentiation rules
Typical course	Multivariable calculus (Calculus III)	Single-variable calculus (Calculus I)

Why It Matters

Partial derivatives are the foundation of multivariable calculus. You will encounter them when computing gradients, finding tangent planes to surfaces, optimizing functions of several variables (like in economics or machine learning), and solving partial differential equations in physics and engineering. Mastering them is essential before studying topics such as the chain rule for multivariable functions, Lagrange multipliers, and vector calculus.

Common Mistakes

Mistake: Treating the other variables as if they also change (differentiating y with respect to x as 1 or dy/dx instead of 0).

Correction: When computing ∂f/∂x, every variable except x is a constant. The derivative of y with respect to x is 0, not 1 and not dy/dx. Only use implicit differentiation (dy/dx) when you are told y depends on x.

Mistake: Using the ordinary derivative symbol d instead of the partial derivative symbol ∂.

Correction: The curly-d notation ∂ signals that other variables are being held constant. Using d can imply a total derivative, which has a different meaning (it accounts for how all variables change). Always use ∂ for partial derivatives.

Related Terms

Derivative — Single-variable version of the same concept
Function — Partial derivatives operate on multivariable functions
Independent Variable — Each partial derivative targets one independent variable
Variable — Partial derivatives distinguish variable from constant
Constant — Other variables are treated as constants
Multivariable Calculus — The branch of calculus where partial derivatives arise
Curly d — The ∂ symbol used in partial derivative notation
Del Operator — Combines partial derivatives into the gradient vector ∇f