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` `what is dy/dx? how to understand it correctly and easily?

8 years ago

Hi,

In calculus, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much a quantity is changing at a given point; for example, the derivative of the position (or distance) of a vehicle with respect to time is the instantaneous velocity (respectively, instantaneous speed) at which the vehicle is traveling.

For a real-valued function of a single real variable, the derivative at a point equals the slope of the tangent line to the graph of the function at that point.

The process of finding a derivative is called differentiation. The fundamental theorem of calculus states that differentiation is the reverse process to integration.Differentiation is a method to compute the rate at which a dependent output y changes with respect to the change in the independent input x. This rate of change is called the derivative of y with respect to x. In more precise language, the dependence of y upon x means that y is a function of x. If x and y are real numbers, and if the graph of y is plotted against x, the derivative measures the slope of this graph at each point. This functional relationship is often denoted y = ƒ(x), where ƒ denotes the function.

The simplest case is when

yis a linear ofx, meaning that the graph ofyagainstxis a straight line. In this case,y=ƒ(x) =mx+c, for real numbersmandc, and the slopemis given by

where the symbol Δ (the uppercase form of the Greek letter Delta) is an abbreviation for "change in." This formula is true because

*y*+ Δ*y*=*ƒ*(*x*+ Δ*x*) =*m*(*x*+ Δ*x*) +*c*=*m**x*+*c*+*m*Δ*x*=*y*+*m*Δ*x*.

It follows that Δ*y* = *m* Δ*x*.

This gives an exact value for the slope of a straight line. If the function *ƒ* is not linear (i.e. its graph is not a straight line), however, then the change in *y* divided by the change in *x* varies: differentiation is a method to find an exact value for this rate of change at any given value of *x*.

Rate of change as a limiting value

such an infinitesimal change in *x* is denoted by *dx*, and the derivative of *y* with respect to *x* is written

suggesting the ratio of two infinitesimal quantities.

8 years ago

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