{"id":6531,"date":"2021-09-27T14:38:32","date_gmt":"2021-09-27T09:08:32","guid":{"rendered":"https:\/\/www.goseeko.com\/blog\/?p=5709"},"modified":"2025-11-09T22:43:04","modified_gmt":"2025-11-09T17:13:04","slug":"what-is-partial-differentiation-2","status":"publish","type":"post","link":"https:\/\/www.goseeko.com\/blog\/what-is-partial-differentiation-2\/","title":{"rendered":"What are partial differentiation?"},"content":{"rendered":"\n

Overview<\/strong><\/h2>\n\n\n\n

When we find the area of a triangle then it depends on base and height, hence we can say that area of the triangle is the function of base and height. In other words, when we have a mathematical function having more than one variables, we use partial differentiation<\/a>.<\/p>\n\n\n\n

Symbolically and mathematically the relationship is defined as below<\/p>\n\n\n\n

z is called a function of two variables x and y if z has one definite value for every pair of values of x and y.<\/p>\n\n\n\n

it is written as<\/p>\n\n\n\n

z = f (x, y)<\/p>\n\n\n\n

The variable x and y are called independent variables while z is called the dependent variable.<\/p>\n\n\n\n

Note- the function z = f (x, y) represents a surface.<\/p>\n\n\n\n

Partial derivatives<\/strong><\/h2>\n\n\n\n

First order partial differentiation<\/a>–<\/strong><\/p>\n\n\n\n

Let f(x , y) be a function of two variables. Then the partial derivative<\/a> of this function with respect to x can be written as and defined as follows:<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Now the partial derivative of f with respect to f can be written as and defined as follows:<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Note: a. while calculating partial derivatives treat all independent variables, other than the variable with respect to which we are differentiating , as constant.<\/p>\n\n\n\n

b. we apply all differentiation rules.<\/p>\n\n\n\n

Higher order partial differentiation-<\/strong><\/p>\n\n\n\n

Let f(x , y) be a function of two variables. Then its second-order partial derivatives, third order partial derivatives and so on are referred as higher order partial derivatives.<\/p>\n\n\n\n

These are second order four partial derivatives:<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

b and c are known as mixed partial derivatives.<\/p>\n\n\n\n

Similarly we can find the other higher order derivatives.<\/p>\n\n\n\n

Solved examples<\/strong><\/h2>\n\n\n\n

Example-1: Calculate <\/strong><\/strong>for the following function<\/strong><\/p>\n\n\n\n

f(x , y) = 3x<\/strong>\u00b3<\/strong>-5y<\/strong>\u00b2<\/strong>+2xy-8x+4y-20<\/strong><\/p>\n\n\n\n

Sol. <\/strong>To calculate the variable y as a constant, then differentiate f(x,y) with respect to x by using differentiation rules,<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Similarly partial derivative of f(x,y) with respect to y is:<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Example-2: Calculate <\/strong><\/strong>for the following function<\/strong><\/p>\n\n\n\n

f( x, y) = sin(y<\/strong>\u00b2<\/strong>x + 5x \u2013 8)<\/strong><\/p>\n\n\n\n

Sol. <\/strong>To calculate the variable y as a constant, then differentiate f(x,y) with respect to x by using differentiation rules,<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Similarly the partial derivative of f(x,y) with respect to y is,<\/p>\n\n\n\n

 [sin(y\u00b2x + 5x \u2013 8)]<\/p>\n\n\n\n

= cos(y\u00b2x + 5x \u2013 8) (y\u00b2x + 5x \u2013 8)<\/p>\n\n\n\n

= 2xycos(y\u00b2x + 5x \u2013 8)<\/p>\n\n\n\n

Interested in learning about similar topics? Here are a few hand-picked blogs for you!<\/p>\n\n\n\n