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Rules for Calculating Derivatives and Differentials for Functions of Multiple Variables

1. Just like the Derivatives calculate the Instantaneous Rate of Change and Differentials calculate the Change in the Value of Functions of a Single Variable, the same is done for Functions of Multiple Variables with the help of Partial Derivatives, Total Derivatives, Partial Differentials and Total Differentials. The following points explain these terms for a Function of Multiple Variables \(F\) given as follows
   
   \(F=f(x_1, x_2, \cdots, x_n)\)   ...(1)
2. Partial Derivatives: The Partial Derivatives of the Function \(F\) as given in equation (1) with respect to Variables \(x_1, x_2, \cdots, x_n\) denoted by \(F_{x_1}', F_{x_2}', \cdots, F_{x_n}'\) is given as
   
   \(F_{x_1}'={\Large\frac{\partial F}{\partial {x_1}}}={\Large \frac{\partial f(x_1, x_2, \cdots, x_n)}{\partial {x_1}}}\)
   
   \(F_{x_2}'={\Large\frac{\partial F}{\partial {x_2}}}={\Large\frac{\partial f(x_1, x_2, \cdots, x_n)}{\partial {x_2}}}\)
   
   \(\vdots\)
   
   \(F_{x_n}'={\Large\frac{\partial F}{\partial {x_n}}}={\Large\frac{\partial f(x_1, x_2, \cdots, x_n)}{\partial {x_n}}}\)
   
   Partial Derivative of any Function of Multiple Variables with respect to One Particular Variable calculates the Instantaneous Rate of Change of the Function when the Value of that Particular Variable changes Infinitesmally and is calculated by Calculating the Derivative of the Function with respect to that Variable while considering the Values of other Variables to be Constant.
   
   For example, the Partial Derivative \(F_{x_1}'\) is calculated by Calculating the Derivative of Function \(F\) with respect to Variable \(x_1\) while considering the Values of other Variables to be Constant.
   
   Similarly the Partial Derivative \(F_{x_2}'\) is calculated by Calculating the Derivative of Function \(F\) with respect to Variable \(x_2\) while considering the Values of other Variables to be Constant and so on.
   
   Please note that each of these Partial Derivatives are calculated using the Rules for Calculating Derivatives for Functions of a Single Variable.
3. Partial Differentials and Total Differentials: The Partial Differentials of the Function \(F\) as given in equation (1) with respect to Variables \(x_1, x_2, \cdots, x_n\) denoted by \(dF_{x_1}, dF_{x_2}, \cdots, dF_{x_n}\) is calculated by Multiplying the Partial Derivatives with the Differentials of the corresponding Variables as given in the following
   
   \(dF_{x_1}=F_{x_1}'dx_1={\Large \frac{\partial F}{\partial {x_1}}}dx_1={\Large \frac{\partial f(x_1, x_2, \cdots, x_n)}{\partial {x_1}}}dx_1\)
   
   \(dF_{x_2}=F_{x_2}'dx_2={\Large \frac{\partial F}{\partial {x_2}}}dx_2={\Large \frac{\partial f(x_1, x_2, \cdots, x_n)}{\partial {x_2}}}dx_2\)
   
   \(\vdots\)
   
   \(dF_{x_n}=F_{x_n}'dx_n={\Large \frac{\partial F}{\partial {x_n}}}dx_n={\Large \frac{\partial f(x_1, x_2, \cdots, x_n)}{\partial {x_n}}}dx_n\)
   
   The Total Differential of the Function \(F\) denoted by \(dF\) is calculated as a Sum of All Partial Differentials as given in the following
   
   \(dF=dF_{x_1}+dF_{x_2}+\cdots+dF_{x_n}=F_{x_1}'dx_1 + F_{x_2}'dx_2+\cdots+F_{x_n}'dx_n={\Large\frac{\partial F}{\partial {x_1}}}dx_1 + {\Large\frac{\partial F}{\partial {x_2}}}dx_2 + \cdots + {\Large\frac{\partial F}{\partial {x_n}}}dx_n\)   ...(2)
4. Implicit Function Rule: If the Function \(F\) is an Implicit Function given as follows
   
   \(F=f(x_1, x_2, \cdots, x_n)=0\)   ...(3)
   
   then the Total Differential of the Function \(F\) denoted by \(dF\) calculated as a Sum of All Partial Differentials is given as following
   
   \(dF={\Large \frac{\partial F}{\partial {x_1}}}dx_1 + {\Large\frac{\partial F}{\partial {x_2}}}dx_2 + \cdots + {\Large\frac{\partial F}{\partial {x_n}}}dx_n=0\)   ...(4)
   
   Since the Total Differential of the Function \(F\) is 0, it allows us to calculate the Derivative of any One Variable with respect to any other Variable if we consider Variables other than the ones which are involved in calculating Derivatives are Constants.
   
   For example, if we want to calculate the Derivative of Variable \(x_1\) with respect to Variable \(x_2\), then we can consider Variables \(x_3, x_4, \cdots, x_n\) to be Constants, which means the Differentials of these Variables \(dx_3, dx_4, \cdots, dx_n\) are 0. Hence equation (4) becomes,
   
   \({\Large \frac{\partial F}{\partial {x_1}}}dx_1 + {\Large \frac{\partial F}{\partial {x_2}}}dx_2 =0\)
   
   \(\Rightarrow {\Large \frac{\partial F}{\partial {x_1}}}dx_1 = - {\Large \frac{\partial F}{\partial {x_2}}}dx_2\)
   
   \(\Rightarrow {\Large \frac{dx_1}{dx_2}} = - {\Large \frac{\frac{\partial F}{\partial {x_2}}} {\frac{\partial F}{\partial {x_1}}}}\)   ...(5)
5. Chain Rule for Function of Multiple Variables: Partial Derivatives/Total Derivative can be calculated using Chain Rule when the Variables of the Function are themselves Functions of One or More Variables. For example, if the Variables \(x_1, x_2, \cdots, x_n\) of Multi Variable Function \(F\) as given in equation (1) are themselves Functions of a Variables \(t_1, t_2,\cdots, t_k\) as follows
   
   \(x_1=f_1(t_1, t_2, \cdots, t_k)\),   \(x_2=f_2(t_1, t_2, \cdots, t_k)\),   \(\cdots\),   \(x_n=f_n(t_1, t_2, \cdots, t_k)\)
   
   This means Partial Derivates of Function F can be Calulated at 2 Level of Variables
   
   At First Level are the Variables \(x_1, x_2, \cdots, x_n\) which are Dependent on Variables \(t_1, t_2,\cdots, t_k\) (Hence these Variables are called Dependent Variables)
   
   At Second Level are the Variables \(t_1, t_2,\cdots, t_k\) which are Indepedent Variables.
   
   In such case the Partial Derivatives of Function \(F\) with respect to Variables \(t_1, t_2,\cdots, t_k\) are given using Chain Rule as
   
   \({\Large \frac{\partial F}{\partial t_1}}={\Large \frac{\partial x_1}{\partial t_1}\frac{\partial F}{\partial x_1}} + {\Large \frac{\partial x_2}{\partial t_1}\frac{\partial F}{\partial x_2}} + \cdots + {\Large \frac{\partial x_n}{\partial t_1}\frac{\partial F}{\partial x_n}}\)
   
   \({\Large \frac{\partial F}{\partial t_2}}={\Large \frac{\partial x_1}{\partial t_2}\frac{\partial F}{\partial x_1}} + {\Large \frac{\partial x_2}{\partial t_2}\frac{\partial F}{\partial x_2}} + \cdots + \frac{\partial x_n}{\partial t_2}{\Large \frac{\partial F}{\partial x_n}}\)
   
   \(\vdots\)
   
   \({\Large \frac{\partial F}{\partial t_k}}={\Large \frac{\partial x_1}{\partial t_k}\frac{\partial F}{\partial x_1}} + {\Large \frac{\partial x_2}{\partial t_k}\frac{\partial F}{\partial x_2}} + \cdots + {\Large \frac{\partial x_n}{\partial t_k}\frac{\partial F}{\partial x_n}}\)
   
   The above can be represented in form of a Chain Rule Matrix Equation as follows
   
   \(\begin{bmatrix} {\Large \frac{\partial F}{\partial t_1}} \\ {\Large \frac{\partial F}{\partial t_2}} \\ \vdots \\ {\Large \frac{\partial F}{\partial t_k}}\end{bmatrix}\)= \(\begin{bmatrix} {\Large \frac{\partial x_1}{\partial t_1}} & {\Large \frac{\partial x_2}{\partial t_1}} & \cdots & {\Large \frac{\partial x_n}{\partial t_1}} \\ {\Large \frac{\partial x_1}{\partial t_2}} & {\Large \frac{\partial x_2}{\partial t_2}} & \cdots & {\Large \frac{\partial x_n}{\partial t_2}} \\ \vdots & \vdots & \ddots & \vdots \\ {\Large \frac{\partial x_1}{\partial t_k}} & {\Large \frac{\partial x_2}{\partial t_k}} & \cdots & {\Large \frac{\partial x_n}{\partial t_k}} \end{bmatrix} \begin{bmatrix} {\Large \frac{\partial F}{\partial x_1}} \\ {\Large \frac{\partial F}{\partial x_2}} \\ \vdots \\ {\Large \frac{\partial F}{\partial x_n}}\end{bmatrix}\)    ...(6)
   
   As can be seen from the Chain Rule Matrix Equation (6) above, the Vector containing Partial Derivatives of Function \(F\) with respect to Independent Variables \(t_1, t_2,\cdots, t_k\) can be represented as a Product of Vector of Partial Derivatives of Function \(F\) with respect to Dependent Variables \(x_1, x_2, \cdots, x_n\) and the Transpose of Jacobian Matrix Partial Derivatives of Dependent Variables with respect to Independent Variables.
   
   Such Chain Rule Matrix Equations can be used to represent Partial Derivatives of Function \(F\) for Any Arbitrary Level of Dependency of Variables. For example, if the Variables \(t_1, t_2,\cdots, t_k\) are themselves Function of Variables \(u_1, u_2,\cdots, u_m\) such that
   
   \(t_1=f_1(u_1, u_2, \cdots, u_m)\),   \(t_2=f_2(u_1, u_2, \cdots, u_m)\),   \(\cdots\),   \(t_k=f_k(u_1, u_2, \cdots, u_m)\)
   
   then the Partial Derivatives of Function \(F\) with respect to Variables \(u_1, u_2,\cdots, u_m\) can be represented using Chain Rule Matrix Equation as
   
   \(\begin{bmatrix} {\Large \frac{\partial F}{\partial u_1}} \\ {\Large \frac{\partial F}{\partial u_2}} \\ \vdots \\ {\Large \frac{\partial F}{\partial u_m}}\end{bmatrix}\)= \(\begin{bmatrix} {\Large \frac{\partial t_1}{\partial u_1}} & {\Large \frac{\partial t_2}{\partial u_1}} & \cdots & {\Large \frac{\partial t_k}{\partial u_1}} \\ {\Large \frac{\partial t_1}{\partial u_2}} & {\Large \frac{\partial t_2}{\partial u_2}} & \cdots & {\Large \frac{\partial t_k}{\partial u_2}} \\ \vdots & \vdots & \ddots & \vdots \\ {\Large \frac{\partial t_1}{\partial u_m}} & {\Large \frac{\partial t_2}{\partial u_m}} & \cdots & {\Large \frac{\partial t_k}{\partial u_m}} \end{bmatrix}\) \(\begin{bmatrix} {\Large \frac{\partial x_1}{\partial t_1}} & {\Large \frac{\partial x_2}{\partial t_1}} & \cdots & {\Large \frac{\partial x_n}{\partial t_1}} \\ {\Large \frac{\partial x_1}{\partial t_2}} & {\Large \frac{\partial x_2}{\partial t_2}} & \cdots & {\Large \frac{\partial x_n}{\partial t_2}} \\ \vdots & \vdots & \ddots & \vdots \\ {\Large \frac{\partial x_1}{\partial t_k}} & {\Large \frac{\partial x_2}{\partial t_k}} & \cdots & {\Large \frac{\partial x_n}{\partial t_k}} \end{bmatrix}\) \(\begin{bmatrix} {\Large \frac{\partial F}{\partial x_1}} \\ {\Large \frac{\partial F}{\partial x_2}} \\ \vdots \\ {\Large \frac{\partial F}{\partial x_n}}\end{bmatrix}\)    ...(7)
   
   If a Function has Multiple Levels of Dependency of Variables, and the Last Level consists of only a Single Variable, then the Derivative of the Function with respect to Variable at the Last Level is called the Total Derivative.
   
   For example, if the Variables \(x_1, x_2, \cdots, x_n\) of Multi Variable Function \(F\) as given in equation (1) are themselves Functions of a Single Variable \(t\) as follows
   
   \(x_1=f_1(t)\),   \(x_2=f_2(t)\),   \(\cdots\),   \(x_n=f_n(t)\)
   
   then the Total Derivative of Function \(F\) with repect to Variable \(t\) is given using Chain Rule as
   
   \({\Large\frac{dF}{dt}}={\Large \frac{d x_1}{dt}\frac{\partial F}{\partial x_1}} + {\Large\frac{d x_2}{dt}\frac{\partial F}{\partial x_2}} + \cdots + {\Large\frac{d x_n}{dt}\frac{\partial F}{\partial x_n}}\)   ...(8)
   
   The above can also be represented in form of a Chain Rule Matrix Equation as follows
   
   \({\Large\frac{dF}{dt}} = \begin{bmatrix} {\Large \frac{d x_1}{dt}} & {\Large \frac{d x_2}{dt}} & \cdots & {\Large \frac{d x_n}{dt}}\end{bmatrix} \begin{bmatrix} {\Large \frac{\partial F}{\partial x_1}} \\ {\Large \frac{\partial F}{\partial x_2}} \\ \vdots \\ {\Large \frac{\partial F}{\partial x_n}}\end{bmatrix}\)   ...(9)