# 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 $f(x,y)$ 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**: $\frac{\mathrm{\partial}}{\mathrm{\partial}x}(uv)=u\frac{\mathrm{\partial}v}{\mathrm{\partial}x}+v\frac{\mathrm{\partial}u}{\mathrm{\partial}x}$**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:

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

### Example:

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

**Partial derivative with respect to $x$**:- $\frac{\mathrm{\partial}f}{\mathrm{\partial}x}=2xy+y\mathrm{cos}(xy)$

**Partial derivative with respect to $y$**:- $\frac{\mathrm{\partial}f}{\mathrm{\partial}y}={x}^{2}+x\mathrm{cos}(xy)$

### 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.