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Also known as: Gradient component
The derivative of a multivariable function with respect to one variable while holding the others constant.
DESCRIPTION
Partial derivatives quantify how a function changes along one coordinate direction in multivariable calculus. They form components of the gradient vector and appear in Jacobians and Hessians. In machine learning, they drive optimization via backpropagation and gradient-based methods. Partial derivatives underpin sensitivity analysis and Taylor approximations. They require differentiability and are interpreted with other variables fixed. In physics and PDEs, they describe spatial and temporal rates of change.
ALTERNATIVE DEFINITIONS
Wikipedia
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant.
Emphasizes holding all other variables constant.
Encyclopaedia Britannica
Partial derivative, in differential calculus, the derivative of a function of two or more variables with respect to one variable, the other variables being kept constant.
Classical calculus framing with multiple variables.
Wolfram MathWorld
The partial derivative is the derivative with respect to one variable of a multivariable function, with all other variables treated as constants.
Concise formal definition used in mathematical references.
RELATED TERMS
Gradient Jacobian Hessian Chain Rule Optimization Gradient Descent Multivariable Calculus Taylor Series
COMPETING TERMS
Total Derivative Directional Derivative
KNOWLEDGE AREAS
TARGET AUDIENCE
Data Scientist AI Engineer Data Analyst
EXTERNAL SOURCES
- Wikipedia - Partial derivative
- MIT OpenCourseWare - Multivariable calculus course on partial derivatives
- Khan Academy - Tutorials and videos on partial derivatives
- Paul’s Online Math Notes - Worked examples of partial derivatives
TOP QUESTIONS
Why are Partial Derivatives important in machine learning?
They are the basis of gradients used in optimization algorithms like gradient descent and backpropagation.
Category: Relevancy
How do Partial Derivatives relate to the Gradient?
The gradient is the vector of all partial derivatives of a scalar function.
Category: Definition
What assumptions are needed to compute a Partial Derivative?
Differentiability with respect to the variable in question and holding others constant are required.
Category: Requirements
What’s a common mistake with Partial Derivatives?
Confusing total and partial derivatives when variables are interdependent via implicit relationships.
Category: Mistakes
How are Partial Derivatives computed symbolically?
Apply univariate differentiation rules to the chosen variable, treating others as constants.
Category: Example
What is the geometric intuition of a Partial Derivative?
It is the slope of the function along one axis direction on the surface.
Category: Example
Do Partial Derivatives commute?
Under smoothness conditions (Clairaut’s theorem), mixed partials are equal.
Category: Concept
Where do Partial Derivatives appear in optimization?
In gradient-based methods, KKT conditions, and second-order methods via Hessians.
Category: Application
How does the Chain Rule involve Partial Derivatives?
It combines partials through composition of multivariable functions to compute derivatives.
Category: Concept