Partial Differentiation

Partial derivatives measure change in one variable while holding others fixed. Learn first and higher-order partials, mixed derivatives and Clairaut's theorem, the multivariable chain rule, implicit differentiation, homogeneous functions and Euler's theorem, and applications to thermodynamics and fields.

How does gas pressure change with temperature at constant volume? How does a beam's stress change along its length at a fixed height? Partial derivatives answer 'what if only this one variable changes?'.

An analogy

Standing on a hillside, ∂h/∂x is the slope if you walk due east; ∂h/∂y is the slope walking due north.

Computing partials

Treat all other variables as constants and differentiate normally. For f = x²y + sin(xy): f_x = 2xy + y cos(xy), f_y = x² + x cos(xy).

Higher and mixed partials

Second partials f_xx, f_yy measure curvature in each direction; mixed f_xy measure twist. For smooth functions the order doesn't matter (Clairaut).

Chain rule

When x and y depend on t (or on other variables u, v), sum the contributions along each path — tree diagrams keep track.

Homogeneous functions and PDEs

Euler's theorem applies to production functions, thermodynamic potentials and scaling laws. Partial derivatives are the building blocks of PDEs (heat, wave, Laplace — M5).

Formulas

• Definition: ∂f/∂x = lim_{h→0} [f(x + h, y) − f(x, y)]/h
• Clairaut: f_xy = f_yx (if continuous)
• Chain rule: dz/dt = (∂z/∂x)(dx/dt) + (∂z/∂y)(dy/dt)
• Implicit: F(x, y) = 0 ⇒ dy/dx = −F_x/F_y
• Euler (homogeneous degree n): x f_x + y f_y = n f

Worked example: Partials and Clairaut

Given: f = x³y² + e^(xy)

Find: f_x, f_y, f_xy.

• f_x = 3x²y² + y e^(xy)
• f_y = 2x³y + x e^(xy)
• f_xy = 6x²y + e^(xy)(1 + xy) = f_yx

Answer: Mixed partials agree

Worked example: Implicit slope

Given: x³ + y³ = 6xy at (3, 3)

Find: dy/dx.

• F_x = 3x² − 6y, F_y = 3y² − 6x
• dy/dx = −(27 − 18)/(27 − 18)

Answer: −1

Common mistakes

• Wrong: Differentiating y when finding ∂/∂x. — Right: Hold y constant.
• Wrong: Forgetting paths in the chain rule. — Right: Sum over all intermediate variables.

Applications

• Thermodynamics: (∂p/∂T)_V, Maxwell relations.
• Heat & fluid flow: Gradients and PDE models.
• Economics of design: Marginal cost of each parameter.

Key takeaways

• Hold other variables constant.
• Mixed partials commute (smooth f).
• Chain rule sums over paths.
• dy/dx = −F_x/F_y implicitly.

Going further

• Jacobian matrices: All first partials of vector functions.
• Hessian matrix: Second partials for optimisation.
• Maxwell relations: Thermodynamic identities from mixed partials.