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Partial Derivative

by Sikander Iqbal
partial derivative

Partial Derivative Definition: 

Partial derivatives are defined as derivatives of a function of multiple variables when all but the variable of interest is held fixed during the differentiation.  

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, using the rules and formulas of differentiation, we obtain what is called the partial derivative of ‘f’ with respect to x which is denoted by 

partial derivative with respect to x

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 

partial derivative with respect to y

Examples: 

Example# 1: 

Find the partial derivatives fx and fy if f(x , y) is given by 

Partial Derivative 1

Solution of example# 1: 

Assume ‘y’ is constant and differentiate with respect to x to obtain 
 

Partial Derivative 2

Now assume x is constant and differentiate with respect to y to obtain 

Partial Derivative 3

Example# 2: 

Find fx and fy if f(x , y) is given by 

Partial Derivative 4

 
Solution to Example 2: 
Differentiate with respect to x assuming y is constant 

Partial Derivative 5

 
Differentiate with respect to y assuming x is constant 

Partial Derivative 6

Example# 3: 

Find fx and fy if f(x , y) is given by 

Partial Derivative 7

Solution to Example 3: 
Differentiate with respect to x assuming y is constant using the product rule of differentiation. 
 

Partial Derivative 8

 
Differentiate with respect to y assuming x is constant. 

Partial Derivative 9

Example# 4: 

Find fx and fy if f(x , y) is given by 

Inserting image...

Solution to Example 4: 
Differentiate with respect to x to obtain 

Inserting image...

 
Differentiate with respect to y 

Partial Derivative 10

Example# 5: 

Find fx(2 , 3) and fy(2 , 3) if f(x , y) is given by 

Partial Derivative 11

 
Solution to Example 5: 
We first find the partial derivatives fx and fy 
fx(x,y) = 2x y 
fy(x,y) = x2 + 2 
We now calculate fx(2 , 3) and fy(2 , 3) by substituting x and y by their given values 
fx(2,3) = 2 (2)(3) = 12 
fy(2,3) = 22 + 2 = 6 

Phrasing and notation 

Here are some of the phrases you might hear in reference to /f operation: 

  • “The partial derivative of  with respect to ” 
  • “Del f, del x” 
  • “Partial f, partial x” 
  • “The partial derivative (of ‘) in the ‘-direction” 

Alternate notation: 

In the same way that people sometimes prefer to write  instead of f / d xwe have the following notation: 

          fx ↔∂x/f  

         fy ↔ ∂f/ y 

       F(some variable)  ↔ ∂f∂ (That same variable

A more formal definition: 

Although thinking of dx or ∂x as really tiny changes in the value of x is a useful intuition, it is healthy to occasionally step back and remember that defining things precisely requires introducing limits. After all, what specific small value would ∂x be? One one hundredth? One one millionth? 10^10^10? 

The point of calculus is that we don’t use any one tiny number, but instead consider all possible values and analyze what tends to happen as they approach a limiting value. The single variable derivative, for example, is defined like this: 

dx/df (x0) = h→0lim       ( h(x0 +h)−f(x0 ))  h 

  • h represents the “tiny value” that we intuitively think of as dx
  • The h→0 under the limit indicates that we care about very small values of h, those approaching 0. 
  • f(x0 +h)−f(x0 ) is the change in the output that results from adding h to the input, which is what we think of as df 

Formally defining the partial derivative looks almost identical. If x,y…), is a function with multiple inputs, here’s how that looks: 

f  (x0 ,y0 ,…)  = h→0lim (h(x0 +hy0  ,…) − (x0 ,y0 ,…))   /h  

The point is that h, which represents a tiny tweak to the input, is added to different input variables depending on which partial derivative we are taking. 

People will often refer to this as the limit definition of a partial derivative. 

You may be interested in Matrix Determinant Calculator.

Second Partial Derivative:  

A brief overview of second partial derivative, the symmetry of mixed partial derivatives, and higher order partial derivatives. 

Notations of Second Order Partial Derivatives: 

 For a two variable function f(x , y), we can define 4 second order partial derivatives along with their notations. 

Partial Derivative 12

Examples with Detailed solutions: 

Example# 1 
Find fxx, fyy given that f(x , y) = sin (x y) 
Solution
fxx may be calculated as follows: 
fxx = ∂2f / ∂x2 = ∂(∂f / ∂x) / ∂x 
= ∂(∂[ sin (x y) ]/ ∂x) / ∂x 
= ∂(y cos (x y) ) / ∂x 
= – y2 sin (x y) ) 
fyy can be calculated as follows: 
fyy = ∂2f / ∂y2 = ∂(∂f / ∂y) / ∂y 
= ∂(∂[ sin (x y) ]/ ∂y) / ∂y 
= ∂(x cos (x y) ) / ∂y 
= – x2 sin (x y)  

Example# 2 
Find fxx, fyy, fxy, fyx given that f(x , y) = x3 + 2 x y. 
Solution
fxx is calculated as follows: 
fxx = ∂2f / ∂x2 = ∂(∂f / ∂x) / ∂x 
= ∂(∂[ x3 + 2 x y ]/ ∂x) / ∂x 
= ∂( 3 x2 + 2 y ) / ∂x 
= 6x 
fyy is calculated as follows: 
fyy = ∂2f / ∂y2 = ∂(∂f / ∂y) / ∂y 
= ∂(∂[ x3 + 2 x y ]/ ∂y) / ∂y 
= ∂( 2x ) / ∂y 
= 0 
fxy is calculated as follows: 
fxy = ∂2f / ∂y∂x = ∂(∂f / ∂x) / ∂y 
= ∂(∂[ x3 + 2 x y ]/ ∂x) / ∂y 
= ∂( 3 x2 + 2 y ) / ∂y 
= 2 
fyx is calculated as follows: 
fyx = ∂2f / ∂x∂y = ∂(∂f / ∂y) / ∂x 
= ∂(∂[ x3 + 2 x y ]/ ∂y) / ∂x 
= ∂( 2x ) / ∂x 
= 2 

Example# 3 
Find fxx, fyy, fxy, fyx given that f(x , y) = x3y4 + x2 y. 
Solution: 
fxx is calculated as follows: 
fxx = ∂2f / ∂x2 = ∂(∂f / ∂x) / ∂x 
= ∂(∂[ x3y4 + x2 y ]/ ∂x) / ∂x 
= ∂( 3 x2y4 + 2 x y) / ∂x 
= 6x y4 + 2y 
fyy is calculated as follows: 
fyy = ∂2f / ∂y2 = ∂(∂f / ∂y) / ∂y 
= ∂(∂[ x3y4 + x2 y ]/ ∂y) / ∂y 
= ∂( 4 x3y3 + x2 ) / ∂y 
= 12 x3y2 
fxy is calculated as follows: 
fxy = ∂2f / ∂y∂x = ∂(∂f / ∂x) / ∂y 
= ∂(∂[ x3y4 + x2 y ]/ ∂x) / ∂y 
= ∂( 3 x2y4 + 2 x y ) / ∂y 
= 12 x2y3 + 2 x 
fyx is calculated as follows: 
fyx = ∂2f / ∂x∂y = ∂(∂f / ∂y) / ∂x 
= ∂(∂[ x3y4 + x2 y ]/ ∂y) / ∂x 
= ∂(4 x3y3 + x2) / ∂x 
= 12 x2y3 + 2x 

The gradient 

The gradient stores all the partial derivative information of a multivariable function.  But it’s more than a mere storage device, it has several wonderful interpretations and many, many uses. 

Partial Derivative 13

Example: 

The gradient of a function f, denoted as ∇f, is the collection of all its partial derivatives into a vector. This is most easily understood with an example. 

Let f(x,y)= x^2y. Find ∇f(3,2). 

Solution:  The gradient is just the vector of partial derivatives. The partial derivatives of f, at the point (x,y)=(3,2) are: 

∂f/∂x (x,y) = 2xy    

∂f/∂x (3,2) = 12 

∂f/∂y (x,y) = x2 

∂f/∂y (3,2) = 9 

Therefore, the gradient is 

∇f(3,2) = 12i+9j = (12,9). 

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