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: The Chain Rule is used for calculating the Total Derivative or Partial Derivatives of any Multi-Variable Function when the Variables of the Function are themselves Functions of One or More Variables. In such cases, Total Derivative and Partial Derivatives are calculated using Total Derivative Operator and Partial Derivative Operators respectively which are themselves calculated using the Chain Rule.
   
   Total Derivative is/can be calculated using Chain Rule under the following 2 circumstances
   1. When the Multi Variable Function has One Independent Variable and Other Variables are Functions of the Independent Variable: For example, if the Variables \(x_1, x_2, \cdots, x_n\) of Multi Variable Function \(F\) as given in equation (1) are such that Variable \(x_1\) is Independent Variable and Variables \(x_2, x_3, \cdots, x_n\) are themselves Functions of a Variable \(x_1\) as follows
      
      \(x_2=f_2(x_1)\),   \(x_3=f_3(x_1)\),   \(\cdots\),   \(x_n=f_n(x_1)\)
      
      then the Total Derivative Operator with respect to Variable \(x_1\) denoted by \({\Large \frac{d}{dx_1}}\) is given using Chain Rule as
      
      \({\Large\frac{d}{dx_1}}={\Large \frac{d x_1}{d x_1}\frac{\partial}{\partial x_1}} + {\Large\frac{d x_2}{d x_1}\frac{\partial}{\partial x_2}} + {\Large\frac{d x_3}{d x_1}\frac{\partial}{\partial x_3}} + \cdots + {\Large\frac{d x_n}{d x_1}\frac{\partial}{\partial x_n}}\)
      
      \(\Rightarrow {\Large\frac{d}{dx_1}}={\Large \frac{\partial}{\partial x_1}} + {\Large\frac{d x_2}{d x_1}\frac{\partial}{\partial x_2}} + {\Large\frac{d x_3}{d x_1}\frac{\partial}{\partial x_3}} + \cdots + {\Large\frac{d x_n}{d x_1}\frac{\partial}{\partial x_n}}\)   ...(6)
      
      Using the Total Derivative Operator as given in equation (6) above, the Total Derivative of Function \(F\) with respect to Variable \(x_1\) is calculated as
      
      \({\Large\frac{dF}{dx_1}}={\Large (\frac{\partial}{\partial x_1}} + {\Large\frac{d x_2}{d x_1}\frac{\partial}{\partial x_2}} + {\Large\frac{d x_3}{d x_1}\frac{\partial}{\partial x_3}} + \cdots + {\Large\frac{d x_n}{d x_1}\frac{\partial}{\partial x_n})}F\)   ...(7)
      
      \(\Rightarrow{\Large\frac{dF}{dx_1}}={\Large \frac{\partial F}{\partial x_1}} + {\Large\frac{d x_2}{d x_1}\frac{\partial F}{\partial x_2}} + {\Large\frac{d x_3}{d x_1}\frac{\partial F}{\partial x_3}} + \cdots + {\Large\frac{d x_n}{d x_1}\frac{\partial F}{\partial x_n}}\)   ...(8)
      
      Please note that the formula of Total Derivative Operator \({\Large \frac{d}{dx_1}}\) given in equation (6) can also be used for calculating the Total Derivatives of Partial Derivatives of \(F\). For example the Total Derivative of \({\Large \frac{\partial F}{\partial x_n}}\) with respect to Variable \(x_1\) can be given as
      
      \({\Large \frac{d}{x_1}(\frac{\partial F}{\partial x_n})}={\Large (\frac{\partial}{\partial x_1}} + {\Large \frac{d x_2}{dx_1}\frac{\partial}{\partial x_2}} + \cdots + {\Large \frac{d x_n}{dx_1}\frac{\partial}{\partial x_n})}{\Large\frac{\partial F}{\partial x_n}}\)   ...(9)
      
      \(\Rightarrow {\Large \frac{d}{x_1}(\frac{\partial F}{\partial x_n})}={\Large \frac{\partial^2 F}{\partial x_1\partial x_n}} + {\Large \frac{d x_2}{dx_1}\frac{\partial^2 F}{\partial x_2\partial x_n}} + \cdots + {\Large \frac{d x_n}{dx_1}\frac{\partial^2 F}{{\partial x_n}^2}}\)   ...(10)
   2. When each Variable of the Multi Variable Function is a Function of a Single Variable: 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 Operator with repect to Variable \(t\) denoted by \({\Large \frac{d}{dt}}\) is given using Chain Rule as
      
      \({\Large\frac{d}{dt}}={\Large \frac{d x_1}{dt}\frac{\partial}{\partial x_1}} + {\Large\frac{d x_2}{dt}\frac{\partial}{\partial x_2}} + \cdots + {\Large\frac{d x_n}{dt}\frac{\partial}{\partial x_n}}\)   ...(11)
      
      Using the Total Derivative Operator as given in equation (11) above, the Total Derivative of Function \(F\) with respect to Variable \(t\) is calculated as
      
      \({\Large \frac{dF}{dt}}={\Large (\frac{d x_1}{dt}\frac{\partial}{\partial x_1}} + {\Large \frac{d x_2}{dt}\frac{\partial}{\partial x_2}} + \cdots + {\Large \frac{d x_n}{dt}\frac{\partial}{\partial x_n})}F\)   ...(12)
      
      \(\Rightarrow {\Large \frac{dF}{dt}}={\Large \frac{d x_1}{dt}\frac{\partial F}{\partial x_1}} + {\Large \frac{d x_2}{dt}\frac{\partial F}{\partial x_2}} + \cdots + {\Large \frac{d x_n}{dt}\frac{\partial F}{\partial x_n}}\)   ...(13)
      
      Please note that the formula of Total Derivative Operator \({\Large \frac{d}{dt}}\) given in equation (11) can also be used for calculating the Total Derivatives of Partial Derivatives of \(F\). For example the Total Derivative of \({\Large \frac{\partial F}{\partial x_n}}\) with respect to Variable \(t\) can be given as
      
      \({\Large \frac{d}{dt}(\frac{\partial F}{\partial x_n})}={\Large (\frac{d x_1}{dt}\frac{\partial}{\partial x_1}} + {\Large \frac{d x_2}{dt}\frac{\partial}{\partial x_2}} + \cdots + {\Large \frac{d x_n}{dt}\frac{\partial}{\partial x_n})}{\Large\frac{\partial F}{\partial x_n}}\)   ...(14)
      
      \(\Rightarrow {\Large \frac{d}{dt}(\frac{\partial F}{\partial x_n})}={\Large \frac{d x_1}{dt}\frac{\partial^2 F}{\partial x_1\partial x_n}} + {\Large \frac{d x_2}{dt}\frac{\partial^2 F}{\partial x_2\partial x_n}} + \cdots + {\Large \frac{d x_n}{dt}\frac{\partial^2 F}{{\partial x_n}^2}}\)   ...(15)
   Partial Derivatives are/can be calculated using Chain Rule when the Variables of the Function are themselves Functions of Multiple Variables. For example, if the Variables \(x_1, x_2, \cdots, x_n\) of Multi Variable Function \(F\) as give in equation (1) are themselves Functions of a Variables \(t_1, t_2,\cdots, t_k\) as follows
   
   \(x_1=f_1(t_1, t_2, \cdots, t_k)\),   \(x_2=f_2(t_1, t_2, \cdots, t_k)\),   \(\cdots\),   \(x_n=f_n(t_1, t_2, \cdots, t_k)\)
   
   then the Partial Derivative Operators with respect to Variables \(t_1, t_2,\cdots, t_k\) are given using Chain Rule as
   
   \({\Large \frac{\partial}{\partial t_1}}={\Large \frac{\partial x_1}{\partial t_1}\frac{\partial}{\partial x_1}} + {\Large \frac{\partial x_2}{\partial t_1}\frac{\partial}{\partial x_2}} + \cdots + {\Large \frac{\partial x_n}{\partial t_1}\frac{\partial}{\partial x_n}}\)
   
   \({\Large \frac{\partial}{\partial t_2}}={\Large \frac{\partial x_1}{\partial t_2}\frac{\partial}{\partial x_1}} + {\Large \frac{\partial x_2}{\partial t_2}\frac{\partial}{\partial x_2}} + \cdots + \frac{\partial x_n}{\partial t_2}{\Large \frac{\partial}{\partial x_n}}\)
   
   \(\vdots\)
   
   \({\Large \frac{\partial}{\partial t_k}}={\Large \frac{\partial x_1}{\partial t_k}\frac{\partial}{\partial x_1}} + {\Large \frac{\partial x_2}{\partial t_k}\frac{\partial}{\partial x_2}} + \cdots + {\Large \frac{\partial x_n}{\partial t_k}\frac{\partial}{\partial x_n}}\)
   
   All the calculations related to Partial Derivative Operators given via Chain Rule can be represented in form of a Matrix Equation as follows
   
   \(\begin{bmatrix} {\Large \frac{\partial}{\partial t_1}} & {\Large \frac{\partial}{\partial t_2}} & \cdots & {\Large \frac{\partial}{\partial t_k}}\end{bmatrix}\)= \(\begin{bmatrix} {\Large \frac{\partial}{\partial x_1}} & {\Large \frac{\partial}{\partial x_2}} & \cdots & {\Large \frac{\partial}{\partial x_n}}\end{bmatrix}\) \(\begin{bmatrix} {\Large \frac{\partial x_1}{\partial t_1}} & {\Large \frac{\partial x_1}{\partial t_2}} & \cdots & {\Large \frac{\partial x_1}{\partial t_k}} \\ {\Large \frac{\partial x_2}{\partial t_1}} & {\Large \frac{\partial x_2}{\partial t_2}} & \cdots & {\Large \frac{\partial x_2}{\partial t_k}} \\ \vdots & \vdots & \ddots & \vdots \\ {\Large \frac{\partial x_n}{\partial t_1}} & {\Large \frac{\partial x_n}{\partial t_2}} & \cdots & {\Large \frac{\partial x_n}{\partial t_k}} \end{bmatrix}\)   ...(16)
   
   As can be seen from the the Matrix Equation (16) above, the Row Matrix containing Partial Derivatives with respect to Independent Variables can be represented as a Product of Row Matrix of Partial Derivatives with respect to Dependent Variables and the Jacobian Matrix of Partial Derivatives of Dependent Variables with respect to Independent Variables.
   
   Such Matrix Equations can be used to represent Partial Derivative Operators for Any Arbitrary Level of Dependency of Variables. For example, if the Variables \(t_1, t_2,\cdots, t_k\) are further 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 Derivative Operators with respect to Variables \(u_1, u_2,\cdots, u_m\) can be represented using Matrix Equation as
   
   \(\begin{bmatrix} {\Large \frac{\partial}{\partial u_1}} & {\Large \frac{\partial}{\partial u_2}} & \cdots & {\Large \frac{\partial}{\partial u_m}}\end{bmatrix}\)= \(\begin{bmatrix} {\Large \frac{\partial}{\partial x_1}} & {\Large \frac{\partial}{\partial x_2}} & \cdots & {\Large \frac{\partial}{\partial x_n}}\end{bmatrix}\) \(\begin{bmatrix} {\Large \frac{\partial x_1}{\partial t_1}} & {\Large \frac{\partial x_1}{\partial t_2}} & \cdots & {\Large \frac{\partial x_1}{\partial t_k}} \\ {\Large \frac{\partial x_2}{\partial t_1}} & {\Large \frac{\partial x_2}{\partial t_2}} & \cdots & {\Large \frac{\partial x_2}{\partial t_k}} \\ \vdots & \vdots & \ddots & \vdots \\ {\Large \frac{\partial x_n}{\partial t_1}} & {\Large \frac{\partial x_n}{\partial t_2}} & \cdots & {\Large \frac{\partial x_n}{\partial t_k}} \end{bmatrix}\) \(\begin{bmatrix} {\Large \frac{\partial t_1}{\partial u_1}} & {\Large \frac{\partial t_1}{\partial u_2}} & \cdots & {\Large \frac{\partial t_1}{\partial u_m}} \\ {\Large \frac{\partial t_2}{\partial u_1}} & {\Large \frac{\partial t_2}{\partial u_2}} & \cdots & {\Large \frac{\partial t_2}{\partial u_m}} \\ \vdots & \vdots & \ddots & \vdots \\ {\Large \frac{\partial t_k}{\partial u_1}} & {\Large \frac{\partial t_k}{\partial u_2}} & \cdots & {\Large \frac{\partial t_k}{\partial u_m}} \end{bmatrix}\)   ...(17)
   
   Once the Partial Derivative Operators are calculated using the Chain Rule, they can be used to calculate the Partial Derivatives of Functions or Partial Derivatives of Partial Derivatives of Functions.