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what is dy/dx? how to understand it correctly and easily?
7 years ago
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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 y is a linear of x, meaning that the graph of y against x is a straight line. In this case, y = ƒ(x) = m x + c, for real numbers m and c, and the slope m is given by
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 y is a linear of x, meaning that the graph of y against x is a straight line. In this case, y = ƒ(x) = m x + c, for real numbers m and c, and the slope m is 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
It follows that Δy = m Δx.
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.
The derivative is the value of the difference quotient as the secant lines approach the tangent line. Formally, the derivative of the function ƒ at a is the limit
of the difference quotient as h approaches zero, if this limit exists. If the limit exists, then ƒ is differentiable at a. Here f′ (a) is one of several common notations for the derivative
Regards,
Rajat
Askiitian Expert
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