# Partial Differentiation

### Partial Differentiation:

• Partial differentiation deals with finding derivatives of functions of multiple variables concerning one variable while keeping the others constant.

### Notation:

• The partial derivative of a function $f\left(x,y\right)$ with respect to $x$ is denoted as $\frac{\mathrm{\partial }f}{\mathrm{\partial }x}$, and with respect to $y$ as $\frac{\mathrm{\partial }f}{\mathrm{\partial }y}$.

### Rules for Partial Differentiation:

• Constant Rule: Derivative of a constant with respect to any variable is always zero.
• Power Rule: Derivative of ${x}^{n}$ with respect to $x$ is $n{x}^{n-1}$, treating other variables as constants.
• Sum and Difference Rule: Derivative of a sum or difference of functions is the sum or difference of their partial derivatives.
• Product Rule: $\frac{\mathrm{\partial }}{\mathrm{\partial }x}\left(uv\right)=u\frac{\mathrm{\partial }v}{\mathrm{\partial }x}+v\frac{\mathrm{\partial }u}{\mathrm{\partial }x}$
• Quotient Rule: $\frac{\mathrm{\partial }}{\mathrm{\partial }x}\left(\frac{u}{v}\right)=\frac{v\frac{\mathrm{\partial }u}{\mathrm{\partial }x}-u\frac{\mathrm{\partial }v}{\mathrm{\partial }x}}{{v}^{2}}$

### Process of Partial Differentiation:

1. Select the variable to differentiate with respect to.
2. Treat other variables as constants and apply standard differentiation rules to find the partial derivative.
3. Repeat the process for each variable of interest, finding all partial derivatives.

### Example:

Given $f\left(x,y\right)={x}^{2}y+\mathrm{sin}\left(xy\right)$, find $\frac{\mathrm{\partial }f}{\mathrm{\partial }x}$ and $\frac{\mathrm{\partial }f}{\mathrm{\partial }y}$.

1. Partial derivative with respect to $x$:
• $\frac{\mathrm{\partial }f}{\mathrm{\partial }x}=2xy+y\mathrm{cos}\left(xy\right)$
2. Partial derivative with respect to $y$:
• $\frac{\mathrm{\partial }f}{\mathrm{\partial }y}={x}^{2}+x\mathrm{cos}\left(xy\right)$

### Applications of Partial Differentiation:

• Physics and Engineering: Used in analyzing functions representing physical phenomena involving multiple variables, such as temperature distribution, fluid flow, and electric fields.
• Economics and Optimization: Applied in optimization problems to find maximum or minimum values of functions involving multiple variables.
• Statistics and Probability: Used in multivariate probability distributions and regression analysis to understand relationships between variables.

### Special Cases:

• Higher Order Partial Derivatives: Involves finding derivatives of partial derivatives, such as $\frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }{x}^{2}}$ or mixed partial derivatives like $\frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }x\mathrm{\partial }y}$.
• Implicit Partial Differentiation: Deriving functions implicitly defined in terms of multiple variables without explicitly solving for one variable.