Derivative of a Function

The limit of the gradient of a chord linking 2 points on a curve as the distance between the points tends to zero; for the function of a single variable, y = f(x), it can be denoted by f ′(x), Df(x), or dy/dx, and is equal to gradient of the curve.

A derivative is instantaneous rate of change of the function. While finding the equation of tangent lines, the slope of curve f(x) at (a, f(a)) is equal to the derivative of f at a. While finding rates of change, the instantaneous rate of change of a function f(x) when x=a is equal to the derivative of f at a.

Derivatives have the notation of their own. The derivative of a function f at a number a can be denoted by f'(a) and can be given by:

So f'(a) represents slope of the tangent line to the curve at a, or the instantaneous rate of change of the function at a. 

Evaluate the derivative function for a given x value, we get a number which is the derivative at a point (which means that the rate of change of f, or the slope of the graph of f ). If y = f (x), an alternate notation for derivative function is dy/dx, which is reminiscent of difference quotient "change in y over the change in x." A third notation uses d/dx as a derivative operator, as in d/dx f (x). This is different from operators like + and -, which take numbers and give a number as the result; d/dx takes a function and gives a new function (the derivative) as a result 

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