Step By Step Calculus » 11.3 - Partial Derivative Rules

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Synopsis
The differentiation rules, in case of the partial derivatives, can be stated as the following.
PD Linearity Rule:\displaystyle \frac{\partial}{\partial x}(cf+g)(x,y)=c\frac{\partial}{\partial x}f(x,y)+\frac{\partial}{\partial x}g(x,y)\displaystyle \frac{\partial}{\partial x}(cf+g)(x,y)=c\frac{\partial}{\partial x}f(x,y)+\frac{\partial}{\partial x}g(x,y).
PD Product Rule:\displaystyle \frac{\partial}{\partial x}(fg)(x,y)=g(x,y)\frac{\partial}{\partial x}f(x,y)+f(x,y)\frac{\partial}{\partial x}g(x,y)\displaystyle \frac{\partial}{\partial x}(fg)(x,y)=g(x,y)\frac{\partial}{\partial x}f(x,y)+f(x,y)\frac{\partial}{\partial x}g(x,y).
PD Reciprocal Rule:\displaystyle \frac{\partial}{\partial x}\left(\frac{1}{g}\right)(x,y)=-\frac{\frac{\partial}{\partial x}g(x,y)}{[g(x,y)]^2},\;\displaystyle \frac{\partial}{\partial x}\left(\frac{1}{g}\right)(x,y)=-\frac{\frac{\partial}{\partial x}g(x,y)}{[g(x,y)]^2},\; provided g(x,y)\neq 0g(x,y)\neq 0.
PD Quotient Rule:\displaystyle \frac{\partial}{\partial x}\left(\frac{f}{g}\right)(x,y)=\frac{g(x,y)\frac{\partial}{\partial x}f(x,y)-f(x,y)\frac{\partial}{\partial x}g(x,y)}{[g(x,y)]^2},\displaystyle \frac{\partial}{\partial x}\left(\frac{f}{g}\right)(x,y)=\frac{g(x,y)\frac{\partial}{\partial x}f(x,y)-f(x,y)\frac{\partial}{\partial x}g(x,y)}{[g(x,y)]^2},
provided g(x,y)\neq 0g(x,y)\neq 0.
PD General Power Rule:\displaystyle \frac{\partial}{\partial x}\left(f^p\right)(x,y)=pf^{p-1}(x,y)\frac{\partial}{\partial x}f(x,y)\displaystyle \frac{\partial}{\partial x}\left(f^p\right)(x,y)=pf^{p-1}(x,y)\frac{\partial}{\partial x}f(x,y) for \forall\;p\in\mathbb{R}\forall\;p\in\mathbb{R} and for all x,yx,y such that f(x,y)> 0f(x,y)> 0 and \dfrac{\partial}{\partial x}f(x,y)\dfrac{\partial}{\partial x}f(x,y) exists.
PD PowEx Rule:\displaystyle \frac{\partial}{\partial x}f^g=gf^{g-1}\frac{\partial f}{\partial x}+f^g\frac{\partial g}{\partial x}\displaystyle \frac{\partial}{\partial x}f^g=gf^{g-1}\frac{\partial f}{\partial x}+f^g\frac{\partial g}{\partial x} for f(x,y)\ge 0f(x,y)\ge 0.
In the above rules, \frac{\partial}{\partial x}\frac{\partial}{\partial x} has been used as an example, \frac{\partial}{\partial y}\frac{\partial}{\partial y} follows the same rules.
The chain rule has several forms for partial derivatives.
  • If h(u)h(u) is a single variable function and u=f(x,y)u=f(x,y) is a two variable function, then \displaystyle \frac{\partial}{\partial x}h(f(x,y)=\frac{d}{du}h(u)\cdot\frac{\partial}{\partial x}f(x,y)\displaystyle \frac{\partial}{\partial x}h(f(x,y)=\frac{d}{du}h(u)\cdot\frac{\partial}{\partial x}f(x,y).
  • If u=f(x,y)u=f(x,y) is a two variable function and x=x(t)x=x(t), y=y(t)y=y(t) are two one variable functions, then \displaystyle \frac{du}{dt}=\frac{\partial f}{\partial x}\cdot \frac{dx}{dt}+\frac{\partial f}{\partial y}\cdot \frac{dy}{dt}\displaystyle \frac{du}{dt}=\frac{\partial f}{\partial x}\cdot \frac{dx}{dt}+\frac{\partial f}{\partial y}\cdot \frac{dy}{dt}.
  • If u=f(x,y)u=f(x,y) is a two variable function and x=g(r,s))x=g(r,s)), y=h(r,s)y=h(r,s) are two 2-variable functions, then \displaystyle \frac{\partial u}{\partial r}=\frac{\partial f}{\partial x}\cdot \frac{\partial g}{\partial r}+\frac{\partial f}{\partial y}\cdot \frac{\partial h}{\partial r}\displaystyle \frac{\partial u}{\partial r}=\frac{\partial f}{\partial x}\cdot \frac{\partial g}{\partial r}+\frac{\partial f}{\partial y}\cdot \frac{\partial h}{\partial r}.