Partial differentiation

Partial differentiation is a cornerstone concept in multivariable calculus, allowing us to study how a function changes with respect to one of its variables while keeping the other variables constant. It generalizes the idea of a derivative from single-variable calculus to functions of multiple variables.

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Definition

If $(f(x, y, z, \dots))$ is a function of several variables, the partial derivative of $(f)$ with respect to a specific variable, say $(x)$ , is defined as:

\frac{\partial f}{\partial x} = \lim_{h \to 0} \frac{f(x + h, y, z, \dots) - f(x, y, z, \dots)}{h}.

Here:

• $(x)$ is treated as a variable, and the other variables $(y, z, \dots)$ are held constant.
• This derivative measures the rate of change of $(f)$ as $(x)$ changes, while all other variables are fixed.

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[!NOTE] Several common notations for partial derivatives include:

• $(\frac{\partial f}{\partial x})$ (most common)
• $(f_{\partial x})$
• $(D_x f)$

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Key Points

1. Conceptual View: Partial derivatives describe how sensitive a function is to changes in one variable, holding others fixed. For example:

   • If $(f(x, y))$ models the temperature of a room where $(x)$ is the horizontal position and $(y)$ is the vertical position, $(\frac{\partial f}{\partial x})$ tells how the temperature changes as you move horizontally.

2. Mathematical Meaning: The partial derivative is essentially the slope of the tangent line to the curve you get when all but one variable is fixed.

3. Higher-Order Derivatives: You can take multiple partial derivatives:

   • Second-order partial derivatives: $(\frac{\partial^2 f}{\partial x^2}$ , $\frac{\partial^2 f}{\partial x \partial y})$ , etc.
   • Mixed derivatives (e.g., $(\frac{\partial^2 f}{\partial x \partial y}))$ are often equal regardless of the order (Clairaut's theorem), provided (f) is sufficiently smooth.

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📒 Example

Let $(f(x, y) = x^2 y + \sin(xy))$ . Compute $(\frac{\partial f}{\partial x})$ and $(\frac{\partial f}{\partial y})$ .

1. Partial Derivative with Respect to $(x)$:

   \frac{\partial f}{\partial x} = \frac{\partial}{\partial x} \big(x^2 y + \sin(xy)\big).

Treat $(y)$ as constant:

   \frac{\partial f}{\partial x} = 2xy + y \cos(xy).

2. Partial Derivative with Respect to $(y)$:

   \frac{\partial f}{\partial y} = \frac{\partial}{\partial y} \big(x^2 y + \sin(xy)\big).

Treat $(x)$ as constant:

   \frac{\partial f}{\partial y} = x^2 + x \cos(xy).

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Applications

Partial differentiation is widely used in:

1. Optimization Problems: To find maxima, minima, and saddle points of multivariable functions.
2. Physics: Calculating gradients, divergence, and curl in vector fields.
3. Economics: Studying how output changes with respect to one input, keeping others constant.
4. Engineering: Analyzing heat flow, stress, and strain in materials.

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