Partial Differentiation
Partial Differentiation
Partial Fractions |
Implicit |
Logarithmic |
Intro Differentiation
Introduction to Partial Differentiation
When a function depends on more than one independent variable, we use partial differentiation to find the rate of change of the function with respect to one variable, while holding the other variables constant.
For a function $z = f(x, y)$:
- The partial derivative of $f$ with respect to $x$ is denoted by $\frac{\partial z}{\partial x}$, $\frac{\partial f}{\partial x}$, $f_x(x,y)$, or $D_x f$. It is found by differentiating $f$ with respect to $x$, treating $y$ as a constant.
- The partial derivative of $f$ with respect to $y$ is denoted by $\frac{\partial z}{\partial y}$, $\frac{\partial f}{\partial y}$, $f_y(x,y)$, or $D_y f$. It is found by differentiating $f$ with respect to $y$, treating $x$ as a constant.
Geometrically, $\frac{\partial f}{\partial x}(x_0, y_0)$ represents the slope of the tangent line to the curve formed by intersecting the surface $z=f(x,y)$ with the plane $y=y_0$ at the point $(x_0, y_0, f(x_0, y_0))$. A similar interpretation holds for $\frac{\partial f}{\partial y}$.
First-Order Partial Derivatives
Calculate First-Order Partial Derivatives
Second-Order Partial Derivatives
Second-order partial derivatives are found by differentiating the first-order partial derivatives again.
For a function $z = f(x, y)$, there are four second-order partial derivatives:
- $\frac{\partial^2 f}{\partial x^2} = \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial x}\right) = f_{xx}$
- $\frac{\partial^2 f}{\partial y^2} = \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial y}\right) = f_{yy}$
- $\frac{\partial^2 f}{\partial y \partial x} = \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial x}\right) = f_{xy}$
- $\frac{\partial^2 f}{\partial x \partial y} = \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial y}\right) = f_{yx}$
Clairaut's Theorem: If $f_{xy}$ and $f_{yx}$ are continuous, then $f_{xy} = f_{yx}$.
Calculate Second-Order Partial Derivatives
Maxima, Minima, and Saddle Points
Partial derivatives are crucial for finding local maxima, local minima, and saddle points of functions of two variables, $z = f(x,y)$.
Critical Points
A point $(a,b)$ in the domain of $f$ is a critical point if both first-order partial derivatives are zero at that point, or if one or both do not exist:
$\qquad f_x(a,b) = 0 \quad \text{and} \quad f_y(a,b) = 0$
Or, $f_x(a,b)$ or $f_y(a,b)$ is undefined.
Local extrema (maxima or minima) can only occur at critical points.
Second Derivative Test
To classify a critical point $(a,b)$ (where $f_x(a,b)=0$ and $f_y(a,b)=0$, and all second partial derivatives are continuous around $(a,b)$), we use the discriminant (or Hessian determinant), $D$:
$\qquad D(a,b) = f_{xx}(a,b) \cdot f_{yy}(a,b) - [f_{xy}(a,b)]^2$
The classification rules are:
- If $D(a,b) > 0$ and $f_{xx}(a,b) > 0$, then $f$ has a local minimum at $(a,b)$.
- If $D(a,b) > 0$ and $f_{xx}(a,b) < 0$, then $f$ has a local maximum at $(a,b)$.
- If $D(a,b) < 0$, then $f$ has a saddle point at $(a,b)$.
- If $D(a,b) = 0$, the test is inconclusive, and other methods are needed to classify the critical point.