Partial derivative

Partial derivative 
Partial derivatives are defined as derivatives of a function of several or multiple variables when all but the variable of interest are held fixed during the differentiation. Most commonly partial derivatives are used in vector calculus and differential geometry.
∂f/∂xm ≡ lim┬(h→0)⁡.  (f(x1,……….xm+h,……….,xn)-f(x1,……….,xm,………..,xn))/h
For brevity, the above mentioned partial derivative is sometimes denoted by fx m.
Partial derivative of a function ‘f’ with respect to the variable ‘x’ is variously given by,
f’x, fx, f,x, ∂xf, or ∂f/∂x
Partial derivative denoted by a symbol of ‘∂’.
If ‘f’ is function of partial derivative with respect to multivariable,
∂^2f/∂x2 = f xx
∂^2f/∂x ∂y = f xy
∂3f/∂x2 ∂y= f xxy
Partial derivative which involved in more than one variable is called as mixed partial derivative. Calculating partial derivative is very simple.
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Partial Derivative Chain Rule
Chain rule is very important rule in partial derivative rules. The chain rule is formula for calculating the derivative of the composition of multivariable functions. For example if ‘f’ and ‘g’ are two functions, then the partial derivative chain rule express the derivative of the composition function ‘f’.’g’ in terms of the derivatives of ‘f’ and ‘g’.
w’(t) = ∂f/∂x dx/dt+  ∂f/∂y   dy/dt

Second Partial Derivative
The differentiations of first order derivative with respect to respective variable, the resulting function are called as second order partial derivative.
Second partial derivative of function of f x with respect to x is,
f xx = (f x)x or ∂^2z/∂x^2 = ∂/∂x   ∂z/∂x   --------------- (1)
Second partial derivative of function of f x with respect to y is,
f xy = (f x)y or ∂^2z/∂y^2 = ∂/∂y ∂z/∂x ----------------- (2)
Second partial derivative of function of f y with respect to y is,
f yy = (f y)y or ∂^2z/∂y^2 = ∂/∂y ∂z/∂y --------------- (3)
The 2nd partial derivative of function of f y with respect to y is,
f yx = (f y)x or ∂^2z/∂x^2 = ∂/∂x ∂z/∂y ------------------ (4)
The equation 1 and 2 are called as direct second partial derivative and the equation 3 and 4 are called as mixed or cross partial derivative.

Find the Indicated Partial Derivative
Find the indicated partial derivative of f(x, y) = y3 x^2, calculate ∂f/∂x (x, y) and calculate ∂f/∂y (x, y)
Let y = b (constant)
Then take partial derivative of ∂f/∂x (x, y) is ordinary derivative of the function g(x) = b3x^2
g(x) = b3x^2
(dg )/dx  = 2b3x
∂f/∂x (x, y) = 2y3x
Let x = b (constant)
Then take partial derivative of ∂f/∂y (x, y) is ordinary derivative of the function g(y) = b3x^2
g(y) = y3b2
(dg )/dy  = 3y^2b2
∂f/∂x (x, y) = 3y^2x^2

Cross Partial Derivative
Cross partial derivative defined as the effect of changing one argument of a function of two or more variables on the derivative of the function with respect to another argument. For example, if y = (x, z), the first derivative of y with respect to ‘x’ is denoted by,
∂y/∂x  = f x
The second order cross-partial derivative is denoted by,
f x z = ∂f