
Tangents, Derivatives and DifferentiationThe DerivativeLet f(x) be a function and assume that for each value of x, we can calculate the slope of the tangent to the graph y = f(x) at x. This slope depends on the value of x that we choose, and so is itself a function. We call this function the derivative of f(x) and denote it by f ´ (x).
Hence, for a function y = f(x), we denote
Note that f(x+h) − f(x) is the change in y corresponding to the change h in x. Leibniz denoted these changes by Δy and Δx respectively. Thus For this reason, we often write. The symbol should be taken as a whole and not treated as “dy divided by dx”. We read it as “deey by deex”. For clarity, we sometimes write or for the derivative of f(x). To compute the derivative of a function f(x) is to differentiate the function f with respect to x. The process of finding a derivative is called differentiation.
Example 1.For a straight line determined by y = mx + c,
the tangent of the graph (line) at each point coincides with the line.
Thus the tangent has a constant slope m. That is, .
We verify this by using the definition of the derivative: Let f(x) = mx + c. Then
Example 2.We find the derivative of f(x) = x^{2}. Hence
Exercise 1.Find the derivative of these functions using the definition of the derivative.
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