Partial Derivatives

Definition

Partial derivatives are a type of derivative that measure the rate of change of a multivariable function with respect to one of its variables, while treating the other variables as constants. They provide a way to analyze the sensitivity of a function to changes in its individual inputs.

5 Must Know Facts For Your Next Test

1. Partial derivatives are used to analyze the behavior of multivariable functions, such as finding the rates of change, tangent planes, and extrema.
2. The notation for a partial derivative of a function $f(x, y)$ with respect to $x$ is $\frac{\partial f}{\partial x}$, which represents the rate of change of $f$ with respect to $x$ while holding $y$ constant.
3. Partial derivatives are a key component in the Chain Rule for differentiating composite functions of multiple variables.
4. Lagrange Multipliers, a method for finding the extrema of a function subject to constraints, relies on partial derivatives to set up the necessary equations.
5. Partial derivatives are used to define the Jacobian matrix, which is essential for transforming integrals between different coordinate systems in multivariable calculus.

Review Questions

• Explain how partial derivatives are used in the context of limits and continuity for multivariable functions.
  • Partial derivatives are crucial in analyzing the limits and continuity of multivariable functions. To determine the limit of a function $f(x, y)$ as $(x, y)$ approaches a point $(x_0, y_0)$, we can examine the partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ at that point. If the partial derivatives exist and are continuous at $(x_0, y_0)$, then the function is continuous at that point. Partial derivatives also play a role in defining the notion of differentiability for multivariable functions, which is a stronger condition than continuity.
• Describe how partial derivatives are used in the context of tangent planes and linear approximations for multivariable functions.
  • Partial derivatives are fundamental in constructing the tangent plane and linear approximation of a multivariable function $f(x, y)$ at a point $(x_0, y_0)$. The tangent plane is given by the equation $z = f(x_0, y_0) + \frac{\partial f}{\partial x}(x_0, y_0)(x - x_0) + \frac{\partial f}{\partial y}(x_0, y_0)(y - y_0)$, where the partial derivatives evaluated at $(x_0, y_0)$ represent the rates of change of the function in the $x$ and $y$ directions, respectively. This tangent plane provides a linear approximation of the function near the point of tangency.
• Analyze how partial derivatives are utilized in the context of the Chain Rule for multivariable functions.
  • The Chain Rule for multivariable functions is a powerful tool that relies on partial derivatives. When a multivariable function $f(x, y)$ is composed with other functions $x(u, v)$ and $y(u, v)$, the Chain Rule states that the partial derivative of $f$ with respect to $u$ is given by $\frac{\partial f}{\partial u} = \frac{\partial f}{\partial x}\frac{\partial x}{\partial u} + \frac{\partial f}{\partial y}\frac{\partial y}{\partial u}$. This formula allows us to differentiate composite multivariable functions by breaking them down into their constituent parts and using the partial derivatives of each component function. The Chain Rule is essential for solving optimization problems and analyzing the sensitivity of multivariable functions to changes in their inputs.

Related terms

Multivariable Function:

A function that depends on more than one independent variable, such as $f(x, y)$ or $g(x, y, z)$.

Total Derivative: The derivative of a multivariable function that considers the changes in all of its independent variables simultaneously.

Gradient:

The vector of partial derivatives of a multivariable function, which points in the direction of the greatest rate of increase of the function.