Undergraduate
  1. Algebra  1.1. Linear Algebra  1.1.1. Matrices  1.1.2. Determinants  1.1.3. Systems of Linear Equations  1.1.4. Vector Spaces  1.1.5. Linear Transformations  1.1.6. Eigenvalues and Eigenvectors  1.1.7. Diagonalization  1.1.8. Inner Product Spaces  1.1.9. Orthogonality and Orthonormal Bases  1.1.10. Singular Value Decomposition  1.2. Abstract Algebra  1.2.1. Groups  1.2.2. Subgroups  1.2.3. Cyclic Groups  1.2.4. Permutation Groups  1.2.5. Homomorphisms  1.2.6. Normal Subgroups  1.2.7. Rings  1.2.8. Fields  1.2.9. Ideals and Quotient Rings  1.2.10. Polynomial Rings  1.2.11. Vector Spaces over Fields  1.2.12. Modules  1.2.13. Galois Theory
  2. Calculus  2.1. Differential Calculus  2.1.1. Limits  2.1.2. Continuity  2.1.3. Derivatives  2.1.4. Chain Rule  2.1.5. Implicit Differentiation  2.1.6. Applications of Derivatives  2.1.7. Optimization Problems  2.1.8. Mean Value Theorem  2.2. Integral Calculus  2.2.1. Definite Integrals  2.2.2. Indefinite Integrals  2.2.3. Techniques of Integration  2.2.4. Improper Integrals  2.2.5. Applications of Integration  2.2.6. Arc Length and Surface Area  2.3. Multivariable Calculus  2.3.1. Partial Derivatives  2.3.2. Gradient  2.3.3. Divergence and Curl  2.3.4. Multiple Integrals  2.3.5. Line Integrals  2.3.6. Surface Integrals  2.3.7. Green's Theorem  2.3.8. Stokes' Theorem  2.3.9. Divergence Theorem  2.3.10. Jacobians
  3. Differential Equations  3.1. Ordinary Differential Equations  3.1.1. First Order Differential Equations  3.1.2. Higher Order Differential Equations  3.1.3. Systems of Differential Equations  3.1.4. Series Solutions  3.1.5. Numerical Methods for ODEs  3.2. Partial Differential Equations  3.2.1. Heat Equation  3.2.2. Wave Equation  3.2.3. Laplace Equation  3.2.4. Separation of Variables  3.2.5. Fourier Transform Methods
  4. Real Analysis  4.1. Sequences and Series  4.1.1. Convergence of Sequences  4.1.2. Series Tests  4.1.3. Power Series  4.1.4. Uniform Convergence  4.2. Functions of Real Variables  4.2.1. Limits and Continuity  4.2.2. Uniform Continuity  4.2.3. Differentiation in Functions of Real Variables  4.2.4. Riemann Integration  4.2.5. Lebesgue Integration
  5. Complex Analysis  5.1. Complex Numbers  5.1.1. Polar Form  5.1.2. Exponential Form  5.2. Functions of a Complex Variable  5.2.1. Analytic Functions  5.2.2. Cauchy-Riemann Equations  5.2.3. Contour Integrals  5.2.4. Cauchy's Integral Theorem  5.2.5. Residue Theorem  5.2.6. Conformal Mapping
  6. Probability and Statistics  6.1. Probability Theory  6.1.1. Basic Probability Concepts  6.1.2. Random Variables  6.1.3. Probability Distributions  6.1.4. Expectation and Variance  6.1.5. Conditional Probability and Bayes' Theorem  6.2. Statistics  6.2.1. Descriptive Statistics  6.2.2. Inferential Statistics  6.2.3. Hypothesis Testing  6.2.4. Regression Analysis  6.2.5. ANOVA
  7. Numerical Methods  7.1. Root-Finding Algorithms  7.2. Numerical Integration  7.3. Numerical Differentiation  7.4. Numerical Solutions to Differential Equations  7.5. Matrix Computations  7.6. Interpolation and Extrapolation
  8. Discrete Mathematics  8.1. Logic and Proof  8.2. Combinatorics  8.3. Graph Theory  8.4. Boolean Algebra  8.5. Recurrence Relations
  9. Linear Programming and Optimization  9.1. Simplex Method  9.2. Duality  9.3. Integer Programming  9.4. Nonlinear Optimization
  10. Topology  10.1. Metric Spaces  10.2. Topological Spaces  10.3. Continuity and Homeomorphisms  10.4. Compactness  10.5. Connectedness
  11. Geometry  11.1. Euclidean Geometry  11.2. Differential Geometry  11.3. Non-Euclidean Geometry  11.4. Projective Geometry
  12. Mathematical Physics  12.1. Fourier Series  12.2. Laplace Transforms  12.3. Applications of Differential Equations  12.4. Special Functions
  13. Set Theory and Logic  13.1. Sets and Subsets  13.2. Relations and Functions  13.3. Cardinality  13.4. Formal Logic  13.5. Zermelo-Fraenkel Set Theory
  14. Functional Analysis  14.1. Normed Spaces  14.2. Banach Spaces  14.3. Hilbert Spaces  14.4. Linear Operators

Undergraduate → Calculus → Multivariable Calculus ↓


Partial Derivatives


Understanding partial derivatives

The partial derivative is a fundamental concept in multivariable calculus, which is the study of functions that involve multiple variables. While ordinary calculus deals with functions of a single variable, multivariable calculus extends those ideas to functions of more than one variable. This extension allows the discovery of a whole new world of mathematical understanding, which is applicable in various scientific fields such as physics, engineering, economics, and others.

Consider a function f(x, y), where the output depends on two input variables, x and y. For a function of two variables, partial derivatives capture how the function changes with respect to each individual variable while holding the other constant.

What is a partial derivative?

The partial derivative of a function with respect to one of its variables measures how the function changes when that specific variable changes, holding all other variables constant. In other words, when you calculate the partial derivative with respect to x, you treat y as a constant and differentiate with respect to x, just as you would in single-variable calculus.

Mathematical notation

There are several notations used for partial derivatives. If f(x, y) is a function, then:

  • The partial derivative of f with respect to x is denoted by ∂f/∂x or fx.
  • The partial derivative with respect to y is ∂f/∂y or fy.

Example calculation

Consider the function f(x, y) = 3x2y + 2y3 - 5x.

To find the partial derivative of f with respect to x, denoted by ∂f/∂x, differentiate it with respect to x:

∂f/∂x = ∂/∂x (3x2y + 2y3 - 5x) = 6xy - 5.

Here, y is considered constant.

For the partial derivative with respect to y, ∂f/∂y is represented by:

∂f/∂y = ∂/∂y (3x2y + 2y3 - 5x) = 3x2 + 6y2.

This time, x is considered a constant.

Visualizing partial derivatives

Visualizing how partial derivatives work can further deepen the understanding. Let's consider a graphical representation.

Suppose the function f(x, y) is represented as a surface in three-dimensional space:

slope along the x-axis slope along the y-axis

In this simple diagram, the blue line shows how the surface slope behaves when you change x variable while keeping y constant, which represents the partial derivative with respect to x. The red line shows a similar concept, but for a change in y while keeping x constant.

Applications of partial derivatives

Gradient vector

An important use of partial derivatives is in creating the gradient vector. For a given function f(x, y, z), the gradient is a vector that points in the direction of the greatest rate of increase of the function, and its magnitude is the rate of increase. The gradient vector is composed of the partial derivatives with respect to each variable:

∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z)

Tangent plane

Partial derivatives are used to determine the equations of tangent planes to surfaces at points. Consider a surface z = f(x, y) The equation of the tangent plane to the surface at the point (x0, y0, z0) is:

z - z0 = (∂f/∂x)x=x0 (x - x0) + (∂f/∂y)y=y0 (y - y0)

Let's take an example function f(x, y) = x2 + y2:

Find the tangent plane at (1, 1, 2).

∂f/∂x = 2x ∂f/∂y = 2y At point (1, 1, 2): 
z - 2 = 2(1)(x - 1) + 2(1)(y - 1)
z - 2 = 2(x - 1) + 2(y - 1)
z = 2x + 2y - 2

Higher-order partial derivatives

Just like in single-variable calculus, we can compute higher-order derivatives in multivariable calculus. Second-order partial derivatives are particularly interesting. They are computed by differentiating the first partial derivative.

  • The second-order partial derivative with respect to x is ∂²f/∂x².
  • Mixed partial derivatives, such as ∂²f/∂x∂y, are calculated by differentiating first relative to y, and then relative to x, or vice versa.

Example of higher-order partial derivatives

A function f(x, y) = x2y + y3 is given:

  • First, calculate the first-order partial derivative:
    • ∂f/∂x = 2xy ∂f/∂y = x2 + 3y2
  • Next, calculate the second-order derivative:
    • ∂²f/∂x² = ∂/∂x (2xy) = 2y
∂²f/∂y² = ∂/∂y (x2 + 3y2) = 6y
∂²f/∂x∂y = ∂/∂y (2xy) = 2x
∂²f/∂y∂x = ∂/∂x (x2 + 3y2) = 2x

Symmetry in mixed partial derivatives

An important theorem in differential calculus is known as the "Clairaut theorem" or the "Schwartz theorem" regarding mixed partial derivatives. It states that if the function and the derivative involved are continuous, then the mixed partial derivatives are equal. In other words:

∂²f/∂x∂y = ∂²f/∂y∂x

Example demonstration

Consider the function f(x, y) = x3y + xy3 Let's show the symmetry of the mixed partial derivatives.

  • First-order partial derivative:
    • ∂f/∂x = 3x2y + y3 ∂f/∂y = x3 + 3xy2
  • Mixed partial derivatives:
    • ∂²f/∂x∂y = ∂/∂y (3x2y + y3) = 3x2 + 3y2 
∂²f/∂y∂x = ∂/∂x (x3 + 3xy2) = 3x2 + 3y2

Thus, prove ∂²f/∂x∂y = ∂²f/∂y∂x for this function.

Conclusion

Partial derivatives are important for understanding how multivariable functions change. They allow us to explore gradients, understand tangent planes, and solve complex differential equations. The symmetry of mixed partial derivatives ensures consistency and adds elegance to mathematical calculations.

This comprehensive guide on partial derivatives will serve as a foundation for further exploration of multivariable calculus topics and their applications in various fields of study.


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Partial Derivatives

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