Partial Derivatives

Definition and Concept

Multivariable Functions

Functions with two or more independent variables: f(x, y, z, ...). Output depends on multiple inputs simultaneously.

Partial Derivative Meaning

Rate of change of f with respect to one variable while holding others constant. Measures sensitivity along one axis.

Formal Limit Definition

Partial derivative of f with respect to x at a point (x₀, y₀):

∂f/∂x (x₀, y₀) = limₕ→0 [f(x₀ + h, y₀) - f(x₀, y₀)] / h

Notation and Symbols

Leibniz Notation

∂f/∂x emphasizes differentiation with respect to x holding others fixed.

Subscript Notation

f_x, f_y denote partial derivatives with respect to x and y respectively.

Differential Operator Notation

∂/∂x acts as an operator on f: (∂/∂x)f.

Computation Techniques

Treat Other Variables as Constants

Differentiate with respect to the chosen variable only, treat remaining variables as constants.

Power Rule and Sum Rule

Apply standard single-variable rules on the variable of differentiation.

Product and Quotient Rules

Apply product/quotient rules treating other variables as constants.

Chain Rule for Composite Functions

Use chain rule if f contains functions of other variables dependent on differentiation variable.

Example

For f(x,y) = x²y + sin(xy):∂f/∂x = 2xy + cos(xy)*y∂f/∂y = x² + cos(xy)*x

Geometric Interpretation

Tangent Planes

Partial derivatives represent slopes of tangent lines to slices of surface parallel to coordinate planes.

Contours and Level Curves

Partial derivatives relate to steepness in directions along coordinate axes on contour plots.

Rate of Change in Directions

Partial derivatives measure instantaneous rate of change along coordinate directions.

Higher-Order Partial Derivatives

Second-Order Derivatives

Partial derivatives applied twice w.r.t same or different variables: ∂²f/∂x², ∂²f/∂x∂y.

Notation

∂²f/∂x² for second derivative w.r.t x; mixed partials like ∂²f/∂x∂y.

Interpretation

Analyze concavity, curvature, and interaction between variables.

Example

f(x,y) = x²y³∂²f/∂x² = 2y³∂²f/∂y² = 6x²y∂²f/∂x∂y = 6xy²

Mixed Partial Derivatives and Clairaut's Theorem

Definition

Mixed partial derivatives: differentiation w.r.t multiple variables in sequence.

Clairaut's Theorem

If f and its partial derivatives are continuous, mixed partials are equal: ∂²f/∂x∂y = ∂²f/∂y∂x.

Conditions for Equality

Continuity and differentiability in domain neighborhood ensure equality.

Importance

Simplifies calculations and confirms smoothness of functions.

Partial Derivatives Chain Rule

Multivariate Composition

For z=f(x,y), where x=g(t), y=h(t), ∂z/∂t involves chain rule.

Formula

dz/dt = (∂f/∂x)(dx/dt) + (∂f/∂y)(dy/dt)

Extension to Multiple Variables

Sum over all paths from independent variables to dependent variable.

Example

f(x,y) = xy, x = t², y = sin tdz/dt = y * 2t + x * cos t = sin t * 2t + t² * cos t

Implicit Differentiation in Multivariable Functions

Implicit Functions

Functions defined by relations involving multiple variables: F(x,y,z) = 0.

Partial Derivative Extraction

Differentiate both sides w.r.t one variable holding others constant, solve for desired partial derivative.

Example

Given x² + y² + z² = 1,∂z/∂x = - (∂F/∂x) / (∂F/∂z) = - (2x) / (2z) = - x/z

Applications

Solve for derivatives when explicit function form unavailable.

Gradient Vector and Directional Derivatives

Gradient Definition

Vector of all first-order partial derivatives: ∇f = (∂f/∂x, ∂f/∂y, ...).

Interpretation

Points in direction of steepest ascent; magnitude equals maximum rate of increase.

Directional Derivative

Rate of change of f in arbitrary direction u: D_uf = ∇f · u.

Formula

D_uf = (∂f/∂x)u_x + (∂f/∂y)u_y + ...

Example

For f(x,y)=x² + y², ∇f = (2x, 2y). Directional derivative at (1,1) along u=(1/√2,1/√2):

D_uf = 2*1*(1/√2) + 2*1*(1/√2) = 2√2

Applications of Partial Derivatives

Optimization

Find local maxima/minima of multivariable functions by setting gradient to zero and analyzing Hessian matrix.

Economics

Marginal cost, revenue functions depend on partial derivatives w.r.t input variables.

Physics

Rate of change of quantities like temperature, pressure in fields.

Engineering

Stress, strain analysis, fluid dynamics, thermodynamics.

Machine Learning

Gradient descent algorithms rely on partial derivatives to update parameters.

Worked Examples

Example 1: Basic Partial Derivatives

Find ∂f/∂x and ∂f/∂y for f(x,y) = 3x²y + 5xy² + 7.

∂f/∂x = 6xy + 5y²∂f/∂y = 3x² + 10xy

Example 2: Higher-Order Partial Derivatives

Find ∂²f/∂x∂y for f(x,y) = e^(xy) + x³y².

∂f/∂x = ye^(xy) + 3x²y²∂²f/∂x∂y = ∂/∂y (∂f/∂x) = e^(xy) + xy e^(xy) + 6x²y

Example 3: Using Chain Rule

Given z = x² + y², x = r cos θ, y = r sin θ. Find ∂z/∂r and ∂z/∂θ.

∂z/∂r = 2x(∂x/∂r) + 2y(∂y/∂r) = 2x cos θ + 2y sin θ = 2r∂z/∂θ = 2x(∂x/∂θ) + 2y(∂y/∂θ) = 2x(-r sin θ) + 2y(r cos θ) = 0

Common Errors and Misconceptions

Mixing Partial and Total Derivatives

Partial derivatives hold other variables constant; total derivatives account for all dependencies.

Ignoring Variable Dependencies

Failing to apply chain rule when variables depend on each other.

Misapplication of Clairaut's Theorem

Assuming mixed partials always equal without checking continuity conditions.

Incorrect Notation Usage

Confusing ∂ (partial) with d (total) derivatives.

Overlooking Domain Restrictions

Partial derivatives may not exist at points with discontinuities or sharp corners.

References

• Stewart, J. "Calculus: Early Transcendentals," 8th Edition, Cengage Learning, 2015, pp. 957-1004.
• Larson, R., & Edwards, B. H. "Calculus," 10th Edition, Brooks Cole, 2009, pp. 850-890.
• Thomas, G. B., & Finney, R. L. "Calculus and Analytic Geometry," 9th Edition, Addison-Wesley, 1996, pp. 721-765.
• Spivak, M. "Calculus on Manifolds," Westview Press, 1965, pp. 20-55.
• Apostol, T. M. "Mathematical Analysis," 2nd Edition, Addison-Wesley, 1974, pp. 180-230.