Can anyone define differentiation in mathematics context?

Answer 1

suppose y = f(x)

any increment in x (delta-x) will have a corresponding change in y (delta-y); hence y+delta-y = f(x+delta-x)

let us write delta-x as d-x and delta-y as d-y for simplicity

then d-y = f(x+d-x) - f(x)

it is elementary so far

now our choice of d-x is arbitrary, right? we can choose any value.... but we are interested in finding out what happens when d-x is tending to zero (well....close and close to zero...)

we consider the rate of change of y with respect to x, or (d-y)/(d-x)

after finding this value, put d-x = 0 and see what happens...

in essence, we have differentiated the function y=f(x)

eg: assume y = x^2

here d-y = (x+d-x)^2 - x^2 = 2x d-x + (d-x)^2

(d-y)/(d-x) = 2x + d-x

as d-x --> 0 we have d-y/d-x = dy/dx = 2x which is the differentiation of the function x^2

for more elaborate concepts, refer to any basic book in calculus

Answer 2

Differentiation is the process of finding the derivative of a function.
A derivative is the instantaneous rate of change of a function. Geometrically the derivate defines the slope of the tangent line to a function at a given point.

Answer 3

The process of computing a derivative. is called differentiation

Answer 4

the limit of the ratio of the change in a function to the corresponding change in its independent variable as the latter change approaches zero

Answer 5

Limit: x->0 (f(a+x)-f(x))/x

Answer 6

slope