Derivatives are defined by taking the limit of the slope of secant lines as they approach a tangent line. Simply put, the derivative of a function will show the slope of the tangent line to any given point x, thus allowing it to be used to calculate other definitions of f(x).

It is hard to directly find the slope of the tangent line to a given function because we only know one point on it, the point where it is tangent to the function. Instead we will approximate the tangent line by secant lines. When we take the limit of the slopes of the nearby secant lines, we will get the slope of the tangent line.

To find the slopes of the nearby secant lines, choose a small number h. h represents a small change in x, and it can be either positive or negative. The slope of the line through the points (x,f(x)) and (x+h,f(x+h)) is

This expression is Newton's difference quotient. The derivative of f at x is the limit of the value of the difference quotient as the secant lines get closer and closer to being a tangent line:

If the derivative of f exists at every point x, we can define the derivative of f to be the function whose value at a point x is the derivative of f at x.

Since immediately substituting 0 for Δx results in division by zero, calculating the derivative directly can be unintuitive. One technique is to simplify the numerator so that the h in the denominator can be cancelled. This happens very easily for polynomials; see calculus with polynomials. For almost all functions, however, the result is a mess. Fortunately there are general rules which make it easy to differentiate most functions that are easy to write down.




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