Difficulty: Intermediate | Prerequisites: Single-variable differentiation (Calculus I), including product rule, quotient rule and chain rule.
Partial derivatives extend the idea of a derivative to functions of two or more variables. Where single-variable calculus asks "how does f change as x changes?", multivariable calculus asks the same question while holding every other input fixed. This is one of the first major concepts in Calculus III, and nearly every topic that follows (gradients, directional derivatives, tangent planes, optimisation) builds on it. If you are comfortable differentiating single-variable functions using the power, product, quotient and chain rules, you have everything you need to start here.
A partial derivative measures how a multivariable function changes when you nudge one input variable while keeping all the others fixed. You compute it exactly like a single-variable derivative, just treating every other variable as a constant. Master this and you have the foundation for gradients, tangent planes, and optimisation in Calculus III.
Partial derivative
The rate of change of a function of several variables with respect to one of those variables, computed while every other variable is held constant. Formally, for f(x, y), the partial derivative with respect to x at a point (a, b) is the limit of [f(a + h, b) - f(a, b)] / h as h approaches 0.
Think of it as: you freeze every input except one, then take the ordinary derivative with respect to that one input.
Subscript notation (f_x, f_y)
A shorthand for the partial derivative of f with respect to x or y. The subscript tells you which variable is changing.
In simple terms, this means: when you see f_x, read it as "the derivative of f treating x as the variable and everything else as a constant."
Limit definition of the partial derivative
The formal expression f_x(x, y) = lim (h to 0) of [f(x + h, y) - f(x, y)] / h. This is the multivariable version of the difference-quotient definition you met in Calculus I.
Think of it as: the same rise-over-run idea from single-variable calculus, but only in the x-direction (or whichever variable you choose).
Leibniz notation (partial f / partial x, partial z / partial x)
The curly-d symbol (the "partial" symbol) replaces the straight d from single-variable calculus to signal that other variables exist and are being held constant.
In simple terms, this means: whenever you see the curly d, you know you are in multivariable territory and other variables are frozen.
When computing the partial derivative with respect to x, treat every other variable (y, z, etc.) as though it were a plain number.
Apply whatever single-variable differentiation rule is needed: power rule, product rule, quotient rule, chain rule.
The process works identically for f_y (freeze x, differentiate with respect to y) and extends to any number of variables.
For z = f(x, y), all of the following mean "the partial derivative of f with respect to x":
f_x
(partial / partial x) f
partial z / partial x
D_x f
They are interchangeable. Your professor or textbook may favour one over the others; be comfortable reading all four.
Given f(x, y) = (x - y) / (x + y), find f_x and f_y.
For f_x, apply the quotient rule treating y as constant:
Top = x - y, so Top_x = 1
Bottom = x + y, so Bottom_x = 1
f_x = [1(x + y) - 1(x - y)] / (x + y)^2 = 2y / (x + y)^2
For f_y, apply the quotient rule treating x as constant:
Top_y = -1, Bottom_y = 1
f_y = [-1(x + y) - 1(x - y)] / (x + y)^2 = -2x / (x + y)^2
Example 1: Polynomial
f(x, y) = x^3 + x^2 y^3 - 2y^2
f_x = 3x^2 + 2x y^3 (treat y^3 and -2y^2 as constants with respect to x)
f_y = 3x^2 y^2 - 4y (treat x^3 as a constant with respect to y)
Example 2: Root and logarithm
f(x, y) = sqrt(x) ln(y)
f_x = ln(y) / (2 sqrt(x)), using the product rule where ln(y) is constant with respect to x
f_y = sqrt(x) / y, using the product rule where sqrt(x) is constant with respect to y
Example 3: Chain rule with trigonometry
f(x, y) = sin^2(2x + 3y)
f_x = 2 sin(2x + 3y) cos(2x + 3y) times 2 = 4 sin(2x + 3y) cos(2x + 3y)
f_y = 2 sin(2x + 3y) cos(2x + 3y) times 3 = 6 sin(2x + 3y) cos(2x + 3y)
The inner derivative with respect to x is 2; with respect to y it is 3. That is the only difference.
Example 4: Three variables with exponentials
f(x, y, z) = x e^(z e^y)
f_x = e^(z e^y), since x appears only as a coefficient
f_y = x e^(z e^y) times z e^y, by the chain rule on the exponent z e^y
f_z = x e^(z e^y) times e^y, by the chain rule on the exponent z e^y
f_x(a, b) = \lim_{h \to 0} \frac{f(a+h,\, b) - f(a,\, b)}{h}f_x(x, y) = \lim_{h \to 0} \frac{f(x+h,\, y) - f(x,\, y)}{h}f_y(x, y) = \lim_{h \to 0} \frac{f(x,\, y+h) - f(x,\, y)}{h}These are the formal limit definitions. In practice you almost never evaluate these limits directly; you use the standard differentiation rules while treating the other variables as constants.
Students often forget to treat other variables as constants and accidentally differentiate them as well. If you are finding f_x, the variable y is just a number; do not apply the power rule or chain rule to it.
When using the quotient rule, students sometimes differentiate the denominator with respect to the wrong variable. Double-check which variable you are differentiating with respect to before you start.
Students sometimes confuse the notation d/dt f(x) with partial f / partial t. If f depends only on x and x does not depend on t, then d/dt of f(x) is simply zero, because x is constant with respect to t. This is the conceptual warmup from the class notes.
Writing the partial symbol as a straight d (df/dx instead of the curly partial f / partial x) is a notational error that can lose marks and, more importantly, signals confusion about whether other variables exist.
Partial derivatives appear in nearly every subsequent topic in Calculus III: gradients, directional derivatives, tangent planes, the chain rule for multivariable functions, and optimisation (Lagrange multipliers). Getting fluent here pays off for the rest of the course.
Expect exam questions that give you a function and ask for all first-order partial derivatives. Polynomial, logarithmic, trigonometric and exponential functions are all fair game.
The quotient rule applied to a two-variable ratio is a favourite exam question because it tests both the quotient rule mechanics and the constant-treatment rule at the same time.
Know all four notation variants. Exams may switch between f_x, partial f / partial x, partial z / partial x, and D_x f within a single problem set.
True or false: When computing f_x for f(x, y) = 5xy^2, the term y^2 is treated as a constant.
True.
Fill in the blank: f_x for f(x, y) = x^3 + x^2 y^3 - 2y^2 is ______.
3x^2 + 2x y^3.
True or false: If f depends only on x, then d/dt of f(x) equals f'(x).
False. If x does not depend on t, then d/dt of f(x) = 0.
Fill in the blank: The four equivalent notations for the partial derivative of f with respect to x are f_x, ______, ______, and D_x f.
(partial / partial x) f and partial z / partial x (where z = f(x, y)).
Q: Find f_x and f_y for f(x, y) = (x - y) / (x + y).
A: f_x = 2y / (x + y)^2, f_y = -2x / (x + y)^2. Apply the quotient rule, treating the other variable as a constant in each case.
Q: Find all first partial derivatives of f(x, y, z) = x e^(z e^y).
A: f_x = e^(z e^y). f_y = x e^(z e^y) times z e^y. f_z = x e^(z e^y) times e^y. The chain rule is needed for f_y and f_z because the exponent z e^y depends on those variables.
Q: Let f(x, y) = sin^2(2x + 3y). Explain why f_y / f_x = 3/2.
A: Both derivatives have the same structure, 2 sin(2x + 3y) cos(2x + 3y), multiplied by the inner derivative. The inner derivative with respect to x is 2 and with respect to y is 3, so the ratio is 3/2.
Q: If f(x) = 3x^2 tan(e^(5x)), what is d/dt of f(x)?
A: Zero. f depends only on x, and x does not depend on t, so f is constant with respect to t.
Partial derivatives are the building block for the gradient vector, which points in the direction of steepest ascent and is central to optimisation and machine learning. They also feed into the multivariable chain rule, directional derivatives, and tangent-plane equations, all of which appear later in MATH 163.
In physics and engineering, partial derivatives describe how a quantity (temperature, pressure, displacement) changes in one spatial direction while others are held fixed. This is how engineers model heat flow, fluid dynamics, and structural stress.
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