Let f(x,y) be a function with two variables. If we keep y constant and differentiate f (assuming f is differentiable) with respect to the variable x, we obtain what is called the partial derivative of f with respect to x which is denoted by
Similarly If we keep x constant and differentiate f (assuming f is differentiable) with respect to the variable y, we obtain what is called the partial derivative of f with respect to y which is denoted by
We might also define partial derivatives of function f as follows:
∂f
∂x

= 
lim h→0 
f(x + h , y)  f(x , y)
h

∂f
∂y

= 
lim k→0 
f(x , y + k)  f(x , y)
k

We now present several examples with detailed solution on how to calculate partial derivatives.
Example 1: Find the partial derivatives f_{x} and f_{y} if f(x , y) is given by
f(x , y) = x^{2} y + 2x + y
Solution to Example 1:
Assume y is constant and differentiate with respect to x to obtain
f_{x}

= 
∂f
∂x

= 
∂
∂x

[ x^{2} y + 2x + y ]

= 
∂
∂x

[ x^{2} y]

+ 
∂
∂x

[ 2 x ] 
+ 
∂
∂x

[ y ] 
= 
[2 x y] + [ 2 ] + [ 0 ] = 2x y + 2 
Now assume x is constant and differentiate with respect to y to obtain
f_{y}

= 
∂f
∂y

= 
∂
∂y

[ x^{2} y + 2x + y ]

= 
∂
∂y

[ x^{2} y]

+ 
∂
∂y

[ 2 x ] 
+ 
∂
∂y

[ y ] 
= 
[ x^{2} ] + [ 0 ] + [ 1 ] = x^{2} + 1

Example 2: Find f_{x} and f_{y} if f(x , y) is given by
f(x , y) = sin(x y) + cos x
Solution to Example 2:
Differentiate with respect to x assuming y is constant
f_{x}

= 
∂f
∂x

= 
∂
∂x

[ sin(x y) + cos x ] 
= 
y cos(x y)  sin x 
Differentiate with respect to y assuming x is constant
f_{y}

= 
∂f
∂y

= 
∂
∂y

[ sin(x y) + cos x ] 
= 
x cos(x y) 
Example 3: Find f_{x} and f_{y} if f(x , y) is given by
f(x , y) = x e^{x y}
Solution to Example 3:
Differentiate with respect to x assuming y is constant
f_{x}

= 
∂f
∂x

= 
∂
∂x

[ x e^{x y} ]

= 
e^{x y} + x y e^{x y}

= (x y + 1)e^{x y}

Differentiate with respect to y
f_{y}

= 
∂f
∂y

= 
∂
∂y

[ x e^{x y} ]

= (x) (x e^{x y})

= x^{2} e^{x y}

Example 4: Find f_{x} and f_{y} if f(x , y) is given by
f(x , y) = ln ( x^{2} + 2 y)
Solution to Example 4:
Differentiate with respect to x to obtain
f_{x}

= 
∂f
∂x

= 
∂
∂x

[ ln ( x^{2} + 2 y) ]

= 
2x
x^{2} + 2 y

Differentiate with respect to y
f_{y}

= 
∂f
∂y

= 
∂
∂y

[ ln ( x^{2} + 2 y) ]

= 
2
x^{2} + 2 y

Example 5: Find f_{x}(2 , 3) and f_{y}(2 , 3) if f(x , y) is given by
f(x , y) = y x^{2} + 2 y
Solution to Example 5:
We first find f_{x} and f_{y}
f_{x}(x,y) = 2x y
f_{y}(x,y) = x^{2} + 2
We now calculate f_{x}(2 , 3) and f_{y}(2 , 3) by substituting x and y by their given values
f_{x}(2,3) = 2 (2)(3) = 12
f_{y}(2,3) = 2^{2} + 2 = 6
Exercise: Find partial derivatives f_{x} and f_{y} of the following functions
1. f(x , y) = x e^{x + y}
2. f(x , y) = ln ( 2 x + y x)
3. f(x , y) = x sin(x  y)
Answer to Above Exercise:
1. f_{x} =(x + 1)e^{x + y} , f_{y} = x e^{x + y}
2. f_{x} = 1 / x , f_{y} = 1 / (y + 2)
3. f_{x} = x cos (x  y) + sin (x  y) , f_{y} = x cos (x  y)
More on partial derivatives and mutlivariable functions.
Multivariable Functions 