Explain Differential Calculus with words!?

Differential Calculus is a generalization of the "slope" concept in algebra. It deals with slopes or *rates* of all kinds. Any time you have a problem that changes values over time, you will often want to know what that rate of change is in any given situation. This rate of change is called the "derivative."

Although this may not seem like that profound or important a notion at first, think of it this way. Dollars per hour (wages), miles per gallon (gas mileage), kilowatts per hour (energy usage), megabits per second (bandwidth), etc. are all rates, or derivatives. To predict future yields or calculate using these rates, you need differential calculus.

The concept is relatively easy. If you place a straight line on the circumference of a circle, it must be a tangent. It is not so easy if the curve is not regular but rather arbitrary. You could place the line at some point on the curve and estimate its position to be exactly tangent. You could then draw a right triangle with three points (one on the straight line and two on the curve). As the triangle is made smaller and smaller, it will force the line to be more exactly a tangent to the curve. As the triangle becomes vanishingly small (by mathematics) the limit of the lines position is in fact a tangent to the curve. Often, the tangent to a curve can represent an instantaneous slope (or ratio) that is useful (to scientists or engineers). For example the slope may represent distance traveled relative to time which of course is velocity. Differentiating such a curve at any given point on the curve could represent the momentary velocity of a racket on a nonlinear path. Differentiating a new curve representing all the velocities (change of velocity with time) represents acceleration (and therefore momentary force due to g's, etc.). The formulas are in books, but it is good to know what it is all about first.

A memorable momento to the love of my life

To my darling Doo . . so beautiful in her 10kb shot

How could such a cartoon be so hot

Calculus differentiates but not a love as ours

Yet we cannot be until my story tells ...

You see things get quicker

With that you can't bicker

But how much quicker, do they get quicker, my love

Tis then that we use a formula above

But loe

The rhyme is a woe

I need to put this formula ....

below

IF one knows that a stone

Will be by itself alone

just x cubed (x^3) yards away

And there it will never stay ... (ok that struggled)

Then how fast is it moving at the mo ... (now I really stretch)

You need to send your mind to fetch

The three from above the ex

And place it before the ..... variable (whoah I could have been banned for that)

And as nothing in your brain is spared

It is moving at 3 x squared (3x^2)

But how much my love

Does this stone get shoved

As it accelerates all the time

We know were it goes

And we know how it froes

But to know this would be sublime ... (oh come on that was a hard one)

By how much speed does it increase

As it moves along

That wonder will never cease

Until we're done with this song

We must now take the 2 from the x

and times three it makes .... SIX

so it accelerates at just 6x

Now which of your problems is next ....

Not good but it might be memorable!! Good luck

Elementary calculus is usually divided into two branches, differential calculus and integral calculus. Differential calculus is the branch of calculus that is based on the determination of the limit of a certain ratio; whereas, integral calculus is based on the determination of the limit of a certain sum.

Differential calculus, or differentiation, is used primarily to determine the slope or steepness of a curve, also called a curve’s derivative. Slope is a rate of change in a curve—a very steep curve is changing very fast—and calculus is used when a curve is very complicated, such as calculating the slope of a mountain or the speed of a roller coaster.

Differential calculus involves any problem that may be graphed when the desired result is a single point on that graph. For example, if a rancher wants to construct a corral with a limited amount of fence, he can vary the lengths of the sides of the corral. Using calculus, he could determine which lengths would enclose the greatest area and make the largest corral. A graph could be drawn using every possible combination of lengths, and the highest point on that graph—the maximum—would indicate the greatest area.

Differential calculus deals with the instantaneous rate of change (or derivative) of a varying quantity. The derivative of a function is representative of an instantaneous rate of change (infinitesimal change) in the function's value—with respect to its parameters (arguments). The derivative of a function f with respect to x is denoted either as f'(x) or df/dx. All applications of differential calculus are concerned with interpretations of the derivative as the slope of the line tangent to the curve at a specific point, or as the rate of change of the dependent variable with respect to the independent variable.

Employing differential calculus provides a method for determining the slope of a line tangent to a curve, rates of change, points moving on a straight line or other curve, and absolute maxima and minima. It is used by the physical and biological sciences, as well as statistical analysis used in business and social studies.

In order to describe differential calculus as the determination of the limit of a certain ratio we must first understand the ratio. Let f(x) = y, where y is the dependent variable that is a function of the independent variable x. If x0 is a value of x defined in the domain identified for x, then y0 = f(x0) is the corresponding value of y. Let h and k be real numbers, and y0 + k = f(x0 + h). So k = f(x0 + h) - f(x0) and k/h = [f(x0 + h) - f(x0)]/h. This ratio is the difference quotient, and equals the tangent of the curve drawn between the points (x0, y0) and (x0 + h, y0 + k). The difference quotient can be thought of as the average rate of change of y = f(x) with respect to x, within the defined interval. If the limit of this ratio k/h exists as h approaches 0, then this limit is called the derivative of y with respect to x, evaluated at x = x0. The limit is written as limh 0 k/h.

This derivative can be interpreted as the slope of the curve y = f(x) at x = x0. It can also be interpreted as the instantaneous rate of change of y with respect to x at x0. Rules and methods developed by this limit process enabled mathematicians to formulate various equations that provide for rapid calculation of the derivatives of various functions. When the derivative of f(x) is found for all values of x, a new function is obtained—which is itself a function of x. The derivative of f'(x) can be found, and is called the second order derivative of y with respect to x.

Another application of differential calculus is Newton's method, an algorithm to find zeros of a function by approximating the function by its tangent.

Differential calculus?

Best thing ever in maths.

Differential means find the difference which means subtract the area under the curve!