wat is meant by differentiation?? wat do we get by differentiating ?

Question

wat is differentiating with respect to a variable like 'x'?

Answer 1

Differentiation expresses the rate at which a quantity, y, changes with respect to the change in another quantity, x, on which it has a functional relationship. Using the symbol Δ (Delta) to refer to change in a quantity, this rate is defined as a limit of difference quotients.
In mathematics, a derivative is the rate of change of a quantity. A derivative is an instantaneous rate of change: calculated at a specific instant rather than as an average over time. The process of finding a derivative is called differentiation. The reverse process is integration. The two processes are the central concepts of calculus and are related via the fundamental theorem of calculus.
Moreover, for a real-valued function of a single real variable, the derivative at a point equals the slope of the tangent to the graph of the function at that point. Derivatives can be used to characterize many properties of a function, including whether and at what rate the function is increasing or decreasing through a value of the function whether and where the function has maximum or minimum values.
The concept of a derivative can be extended to functions of more than one variable (see multivariable calculus), to functions of complex variables (see complex analysis) and to many other cases.

Differentiation has many applications throughout all numerate disciplines. For example, in physics, the derivative of the position of a moving body is its velocity and the second derivative of the body's position is its acceleration. In turn, the 'velocity' of the body in a given direction is its 'speed' in that direction, another derivative. Speed on position-time axes is the (instantaneous rate of) unit change in position of the body per unit change in time.

Answer 2

Although it's not the first thing that pops into your head when you hear the word 'Differentiation', the concept to think of is 'difference' ... as in 'subtraction' or 'change.'

I think this made-up example might illustrate what's happening:

Suppose I set up a long row of cups along a straight walkway, and in each cup I placed a variable amount of pennies. The cups are spaced exactly 1 unit apart. Your job is to walk along and determine how many pennies are in each cup.

So, beginning at 0 (the origin), you see the cup there has 15 pennies. I could mark this information on a graph as the point (0,15). You take one step and now read out the number of pennies in the cup at position 1 ... '3 pennies'. I mark (1,3). And so on. Eventually, you've read out tens or hundreds of the cups' contents, and my graph has lots of points:
(0,15), (1,3), (2,12), (3,1), etc., etc.

'Differentiating with respect to x' in this case is like asking, 'How did the number of pennies change between each cup?' Obviously, the answer to that question changes all the time, but at least the 'step size' remains constant (1), since you were taking steps exactly 1 unit long between cups.

The 'changes' in the y-value would read: -12, 9, -11, etc.

These numbers represent the 'difference' between the points we found, and all these differences together -- when plotted -- yield an entirely different-looking graph: (1,-12), (2,9), (3,-11), etc.

That graph is called 'the derivative' of the first graph. Derivatives, as you can see, essentially show the values of the SLOPE of the original graph. Sure, the slope is changing all the time ... and if I had spread the pennies out in lots more cups (spaced much closer together), the change between steps might have been much smaller ... yet the graph of the derivative (slopes) would still look the same! (Recall that slope is really 'change in y' divided by 'change in x' ... as I spread the pennies out over more cups with closer spacing, both the numerator and the denominator of the slope formula will change!)

Calculus concerns itself all the time with such tiny steps along the graphs of functions. Terms like 'difference', 'differential', 'derivative', 'instantaneous velocity', 'slope of the tangent', etc. all pretty much mean the same thing. Only the steps are allowed to be SO tiny (we say their 'limit is 0' because they are so incredibly close together) that the corresponding slope graph becomes perfectly smooth and accurate (not just a collection of disconnected points as in my example above).

To answer your last question more fully, you CAN discuss differentiation with respect to the 'y' variable as well ... that would be the same as asking, "How far would I have to step to earn just 1 penny?" From the above example, you might guess that stepping 1/15 of a unit works near the origin, but is 1/3 of a unit as you pass x=1, etc.

Unlike algebra, wherein we talked mostly about the slope of a line (which is always constant), calculus doesn't mind at all talking about the changing slopes of non-linear curves. Once you know how to find the derivative (function) of a given function, you can calculate the slope at just about ANY point you want, easily!

Hope this helps!

Answer 3

Differentiation, as in dx/dt, is the change in posistion with respect to time. Mathematically, the derivative of a function, is the slope of the tangent line to that function.
As with position, (x) the differential of the position function (dx/dt) is the velocity, and the differential of velocity (d/dt(dx/dt) is the acceleration. For a good explaination, consult any calculus book or a web search.

Answer 4

In simple, differentiation is to 'difrag' or spread into smaller parts, like a machine removed their nuts and bolts.
Integration is to assemble them up back.

With respect to 'x' means the particular available in the equation that is to differentiate.

As example, your are going to remove specific parts of a machine that has a function of 'x' , maybe function 2 (regulate n cool) the machine down.

Thus, u r removing(differentiate) everything related to that function only.

Answer 5

Answers include, but are not limited to, the following:

(1) The slope of the tangent line to a curve at a given point.

(2) The rate of change of y with respect to x.

(3) The limit, as h approaches 0, of the quantity

(f(x + h) - f(x))/h

where y = f(x).

Answer 6

Several schools of thought here:

1) it is an analytic process, the limit as h -> 0 of the difference quotient [f(x+h)-f(x)]/h

2) it is a geometric process, finding tangents (tangent lines, tangents planes) to curves

3) it is a numerical process, finding the best local linear approximation to the function values

4) it is an algberaic process, with a set of algebraic rules. So differentiation is a function (itself!) that has domain {the set of differentiable functions} and range {the set of derivatives} and it is an algebraic process D that maps f -> f'

Answer 7

For a function of x say f(x), differentiating means finding the equation for the slope of the tangent line to the curve f(x).

Answer 8

let y=f(x) graphically is represent by a curve
the differentiable of f(x) with respect to x is dy/dx
it means the slope of tangent of this curve at the point (x,y)