Mathematics.

multivariable calculus

Partial Derivatives

Calculus III40 minDifficulty6 out of 10

You should know: derivative

Overview

A partial derivative measures how a multivariable function changes as ONE variable changes, holding all others fixed. For a surface z = f(x,y), the partial derivative with respect to x is the slope of the curve you get by slicing the surface with a plane of constant y.

Intuition

Standing on a hillside (a surface z=f(x,y)), the partial derivative ∂f/∂x tells you the steepness if you only walk east-west, ignoring any north-south slope. ∂f/∂y tells you the steepness walking north-south only. Neither one alone describes the full slope in an arbitrary direction — that requires combining both into the gradient — but each partial derivative isolates exactly one direction's worth of change.

Formal Definition

Definition

The partial derivative of f with respect to x, holding y constant:

∂f∂x=lim⁡h→0f(x+h,y)−f(x,y)h\frac{\partial f}{\partial x} = \lim_{h \to 0} \frac{f(x+h, y) - f(x, y)}{h}

Same limit definition as the ordinary derivative, but y is frozen

Notation

NotationMeaning
∂f∂x\frac{\partial f}{\partial x}Partial derivative of f with respect to xAlso written: f_x, ∂_x f
∇f=(∂f∂x,∂f∂y)\nabla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}\right)Gradient — vector of all first partial derivatives

Properties

Mixed partials (Clairaut's theorem)

∂2f∂x∂y=∂2f∂y∂x\frac{\partial^2 f}{\partial x \partial y} = \frac{\partial^2 f}{\partial y \partial x}

Condition: when both are continuous

Sum rule

∂∂x(f+g)=∂f∂x+∂g∂x\frac{\partial}{\partial x}(f+g) = \frac{\partial f}{\partial x} + \frac{\partial g}{\partial x}

Applications

The heat equation, wave equation, and virtually every PDE in physics is built from partial derivatives describing how quantities change in space and time simultaneously.

3D Visualization

Slice the surface to see each partial derivative

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Worked Examples

  1. Treat y as a constant, differentiate with respect to x.

    ∂f∂x=2xy\frac{\partial f}{\partial x} = 2xy

  2. Treat x as a constant, differentiate with respect to y.

    ∂f∂y=x2+3\frac{\partial f}{\partial y} = x^2 + 3

Answer: ∂f/∂x = 2xy, ∂f/∂y = x² + 3

Practice Problems

Difficulty 5/10

Find ∂f/∂x for f(x,y) = sin(xy).

Difficulty 6/10

A heated plate has temperature T(x,y) = 100 − 2x² − y². Find the rate of temperature change in the x-direction at the point (3, 4), and interpret its sign.

Difficulty 6/10

A beam's deflection depends on load P and length L as δ(P, L) = P·L³/(3EI). Find ∂δ/∂P and explain what it represents physically.

Common Mistakes

Common Mistake

Forgetting to treat the other variable as a genuine constant, and accidentally differentiating it too.

∂/∂x[x²y] = 2xy, NOT 2xy + x²(dy/dx) — y is frozen, full stop, unlike implicit differentiation.

Quiz

For f(x,y) = x³y², what is ∂f/∂y?

Flashcards

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Historical Background

Partial derivatives emerged naturally as calculus was extended to physics problems involving several variables in the 18th century — Euler used partial-derivative-style notation for problems in fluid mechanics as early as the 1730s. The modern ∂ notation (a stylized 'd', distinguishing partial from ordinary derivatives) was introduced by Adrien-Marie Legendre in 1786 and later popularized by Carl Gustav Jacobi.

  1. 1730s

    Euler works with multivariable rates of change in fluid mechanics problems

    Leonhard Euler

  2. 1786

    Legendre introduces the ∂ symbol

    Adrien-Marie Legendre

Summary

  • Partial derivative ∂f/∂x measures change in f as x varies, holding other variables constant.
  • Same limit definition as ordinary derivatives, just freezing everything else.
  • The gradient ∇f collects all partial derivatives into one vector, pointing toward steepest increase.
  • Clairaut's theorem: mixed partials commute when continuous.
  • Foundation of gradient descent, PDEs, and multivariable optimization.

References

  1. WebsiteWikipedia — Partial derivative