Partial Derivatives

Definition:

Partial derivatives measure how a multivariable function changes as one of its variables is varied, while keeping the other variables constant. Given a function $f (x_{1}, x_{2}, \dots, x_{n})$ , the partial derivative of $f$ with respect to $x_{i}$ is denoted by:

\frac{\partial f}{\partial x_{i}} or f_{x_{i}}

This notation indicates that we treat all other variables $(x_{1}, x_{2}, \dots, x_{i - 1}, x_{i + 1}, \dots, x_{n})$ as constants and differentiate with respect to $x_{i}$ .

Example:

Consider a function $f (x, y) = 3 x^{2} y + 2 y^{2}$ . The partial derivative of $f$ with respect to $x$ is:

\frac{\partial f}{\partial x} = 6 x y

Here, $y$ is treated as a constant. Similarly, the partial derivative of $f$ with respect to $y$ is:

\frac{\partial f}{\partial y} = 3 x^{2} + 4 y

Geometric Interpretation:

Partial derivatives give us the slope of the function along a specific axis. For a function $f (x, y)$ , the partial derivative $\frac{\partial f}{\partial x}$ represents the rate of change of $f$ with respect to $x$ , while holding $y$ constant. This is equivalent to slicing the surface $z = f (x, y)$ parallel to the $y z$ -plane and examining the slope of the curve along the $x$ -direction.

Similarly, $\frac{\partial f}{\partial y}$ represents the rate of change of $f$ in the $y$ -direction, with $x$ held constant.

Visualizing Partial Derivatives:

Consider a surface $z = f (x, y)$ . The partial derivative $\frac{\partial f}{\partial x}$ corresponds to the slope of the tangent line in the direction of the $x$ -axis at any point $(x, y)$ . Geometrically, this is the slope of the curve you would see if you "cut" the surface along a plane parallel to the $y z$ -plane. Similarly, $\frac{\partial f}{\partial y}$ is the slope of the tangent line in the direction of the $y$ -axis.

Higher-Order Partial Derivatives:

Just like with single-variable functions, we can take higher-order derivatives in multivariable functions. A second-order partial derivative is obtained by differentiating a first-order partial derivative. There are several types of second-order partial derivatives:

$\frac{\partial^{2} f}{\partial x^{2}}$ : The second derivative of $f$ with respect to $x$ twice (also called the second-order derivative in $x$ ).
$\frac{\partial^{2} f}{\partial y^{2}}$ : The second derivative of $f$ with respect to $y$ twice.
$\frac{\partial^{2} f}{\partial x \partial y}$ or $\frac{\partial^{2} f}{\partial y \partial x}$ : Mixed partial derivatives.

Clairaut’s Theorem on Mixed Partial Derivatives:

If the mixed partial derivatives are continuous, then the order of differentiation does not matter. That is:

\frac{\partial^{2} f}{\partial x \partial y} = \frac{\partial^{2} f}{\partial y \partial x}

Example of Higher-Order Partial Derivatives:

For the function $f (x, y) = x^{3} y^{2} + 4 x y$ , the second-order partial derivatives are:

$\frac{\partial^{2} f}{\partial x^{2}} = 6 x y^{2} + 4 y$
$\frac{\partial^{2} f}{\partial y^{2}} = 2 x^{3}$
Mixed partial derivatives: $\frac{\partial^{2} f}{\partial x \partial y} = \frac{\partial^{2}}{\partial x} (3 x^{2} y^{2} + 4 y) = 6 x^{2} y + 4$

Chain Rule for Partial Derivatives:

The chain rule in the context of partial derivatives allows us to differentiate composite functions. Suppose $z = f (x, y)$ and both $x$ and $y$ are functions of another variable $t$ . The chain rule expresses the derivative of $z$ with respect to $t$ as:

\frac{d z}{d t} = \frac{\partial f}{\partial x} \frac{d x}{d t} + \frac{\partial f}{\partial y} \frac{d y}{d t}

This can be generalized for functions of more variables and multiple intermediate dependencies.

Example:

Let $z = x^{2} + x y$ , where $x = t^{2}$ and $y = e^{t}$ . Using the chain rule:

\frac{d z}{d t} = \frac{\partial z}{\partial x} \frac{d x}{d t} + \frac{\partial z}{\partial y} \frac{d y}{d t}

We calculate:

\frac{\partial z}{\partial x} = 2 x + y, \frac{\partial z}{\partial y} = x

Substituting $x = t^{2}$ and $y = e^{t}$ :

\frac{d z}{d t} = (2 t^{2} + e^{t}) (2 t) + (t^{2}) (e^{t})

Applications of Partial Derivatives:

Optimization:

Partial derivatives are essential in finding the local maxima and minima of functions of multiple variables. To find critical points (where the function might have a local max or min), you set the gradient equal to zero:

\nabla f = 0

Solving this system of equations provides the critical points. From there, the second derivative test (involving second-order partial derivatives) can help determine whether the critical point is a local maximum, local minimum, or a saddle point.

Example:

Consider the function $f (x, y) = x^{2} + y^{2} + 2 x - 4 y$ . The partial derivatives are:

\frac{\partial f}{\partial x} = 2 x + 2, \frac{\partial f}{\partial y} = 2 y - 4

Setting both derivatives to zero:

2 x + 2 = 0 and 2 y - 4 = 0

Solving these equations gives the critical point $(x, y) = (- 1, 2)$ .

Tangent Planes:

The equation of a tangent plane to a surface $z = f (x, y)$ at the point $(x_{0}, y_{0}, z_{0})$ can be written as:

z - z_{0} = f_{x} (x_{0}, y_{0}) (x - x_{0}) + f_{y} (x_{0}, y_{0}) (y - y_{0})

This gives a linear approximation of the surface near the point $(x_{0}, y_{0}, z_{0})$ .

Example:

For the function $f (x, y) = x^{2} + y^{2}$ , find the equation of the tangent plane at the point $(1, 1, 2)$ :
The partial derivatives are:

f_{x} (x, y) = 2 x, f_{y} (x, y) = 2 y

At the point $(1, 1)$ :

f_{x} (1, 1) = 2, f_{y} (1, 1) = 2

Thus, the equation of the tangent plane is:

z - 2 = 2 (x - 1) + 2 (y - 1)

Simplifying:

z = 2 x + 2 y - 2