Derivative and Differentiation

What is derivative and differentiation

Derivative is a measure of how a function changes as its input changes. It essentially gives the rate of change or the slope of a function at any given point. Differentiation is a process to find a derivative.

image is from wikipedia "Derivative"

The graph of a function, drawn in black, and a tangent line to that graph, drawn in red. The slope of the tangent line is equal to the derivative of the function at the marked point. - wikipedia "Derivative" page

image is from wikipedia "Derivative"

How differentiation is calculated

Differentiation is calculated as following:

.

If we have this equation, 

the derivative can be calculated by 

.

Please note that the C is a constant number (a number which is not multiplied by the variable x). And constant ignored when the derivative is calculated. So, if we get the derivative of this function

,

its derivative will be:

.

When there is only one variable, it is easy to calculate.

But what if there are multiple variables in a function? In that case, we can find partial derivative or total derivative of the function.

Partial derivative

The partial derivative of a function f(x, y, ....) is denoted variously such as:

,  ,  .

For example, partial derivative of the following function with respect to the variable x 

is calculated by regarding the other variable "y" as constant, such as,

.