Click here to see an animation of the derivative

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1 Differentiation Massoud Malek Derivative Te concept of derivative is at te core of Calculus; It is a very powerful tool for understanding te beavior of matematical functions. It allows us to optimize functions, wic means to find teir maximum or minimum values, as well as to determine oter valuable qualities describing functions. Real-world applications are endless. Te geometrical definition of te derivative of a curve is te slope of te tangent lines at points on te curve. Te pysical definition is te instantaneous rate of cange; for example, te derivative of te position of a moving object wit respect to time is te object s instantaneous velocity. Differentiation A metod of computing te rate at wic a dependent variable y(x) canges wit respect to te cange in te independent variable x. Tis rate of cange is called te derivative of y wit respect to x and is denoted by d y d x or just y (x). If y = f(x) is a function of x, ten te derivative is denoted by d f(x) or just f (x). If te grap of y is plotted against d x x, te derivative measures te slope of te tangent line to tis grap at eac point. Click ere to see an animation of te derivative Te derivative of y = f(x) wit respect to x at a is te slope of te tangent line to te grap of f(x) at a. Te slope of te tangent line is very close to te slope of te secant line troug (a, (a)) and a nearby point on te grap, for example (a +, (a + ))., is very close to zero. Te slope m of te secant line is te difference between te y values of tese

2 Massoud Malek Differentiation points divided by te difference between te x values, tat is, m = f(x) x f(x + ) f(x) =. Tis expression is Newton s difference quotient. Te derivative is te value of te difference quotient as te secant lines approac te tangent line. Formally, te derivative of te function f(x) at a is te limit f (a) f(a + ) f(a) of te difference quotient as approaces zero. If te limit exists, ten f(x) is differentiable at a. Te secant to curve y = (x) passing troug (x, f(x)) and (x+, f(x+)) Te tangent line as limit of secants f (a) may also be defined as: Hence f (a) f(a) f(a ) or f (a) f(a + ) f(a ) f (a) f(a + ) f(a) f(a) f(a ) f(a + ) f(a ) Example Te function f(x) = x, representing a parabola, is differentiable at x =, wit f () = 4. Tis result is establised by calculating te limit as approaces zero of te difference quotient of f() : f () f( + ) f() = 4. ( + ) 4 4 +

3 Massoud Malek Differentiation 3 Continuity and differentiability If y = f(x) is differentiable at a, ten f(x) must also be continuous at a. Consider te non-continuous step function 1 if x < 1, f(x) = 3 if x 1. We ave f(1 + ) f(1) lim Terefore f(x) is not differentiable at x = 1. = 0 and lim f(1) f(1 ) =. Even if a function is continuous at a point, it may not be differentiable tere. For example, te absolute value function f(x) = x is continuous at x = 0, but it is not differentiable tere. Since f(0 + ) f(0) lim = 1 and lim f(0) f(0 ) = 1. In summary: for a function to ave a derivative it is necessary to be continuous, but continuity alone is not sufficient. Higer Derivatives Let f(x) be a differentiable function, and let f (x) be its derivative. Te derivative of f (x) (if it as one) is written f (x) and is called te second derivative of f (x). Similarly, te derivative of a second derivative, if it exists, is written f (x) 0r f (3) (x) and is called te tird derivative of f(x). Tese repeated derivatives are called iger-order derivatives.

4 Massoud Malek Differentiation 4 If a f(x) as a derivative, it may not ave a second or tird derivative. For example, let x, if x 0 f(x) = x, if x 0. Calculation sows tat is a differentiable function wose derivative is x, if x 0 f (x) = x, if x 0. f (x) is twice te absolute value function, and it does not ave a derivative at zero. Similar examples sow tat a function can ave k derivatives for any non-negative integer k, but no (k + 1) order derivative. A function tat as k successive derivatives is called k times differentiable. If in addition te k t derivative is continuous, ten te function is said to be of differentiability class C k. (Tis is a stronger condition tan aving k derivatives. A function tat as infinitely many derivatives is called infinitely differentiable or smoot. Every polynomial function is infinitely differentiable. By standard differentiation rules, if a polynomial of degree n is differentiated n times, ten it becomes a constant function. All of its subsequent derivatives are identically zero. In particular, tey exist, so polynomials are smoot functions. Te derivatives of a function f(x) at a point k provide polynomial approximations to tat function near k. For example, if f(x) is twice differentiable, ten f(x + ) f(x) + f (x) + 1f (x) in te sense tat f(x + ) f(x) f (x) 1 lim f (x) = 0. Critical Point A critical point of a function is a point in te domain, were eiter te function is not differentiable or its derivative is 0. For example, te point x = 0 is a critical point of all tree functions f(x) = x, g(x) = x, and (x) = x. Because f (x) = 0, so te tangent line to te curve of f(x) at x = 0 is a orizontal line; g(x) = x is not differentiable at x = 0; and finally, te tangent line to te curve of (x) at x = 0 is a vertical line. In te case, te derivative is zero; te point is called a stationary point of te function.

5 Massoud Malek Differentiation 5 Inflection point An inflection point or a point of inflection, is a point on a curve at wic te curvature canges sign. Te curve canges from being concave upwards (positive curvature) to concave downwards (negative curvature), or vice versa. If one imagines driving a veicle along a winding road, inflection point is te point at wic te steering-weel is momentarily straigt, wen being turned from left to rigt or vice versa. At an inflection point, te second derivative may be zero, as in te case of te inflection point x = 0 of te function y = x 3, or it may fail to exist, as in te case of te inflection point x = 0 of te function y = x 1/3. Local Maxima and Minima By Fermat s teorem, local maxima and minima of a function can occur only at its critical points. However, not every stationary point is a maximum or a minimum of te function it may also correspond to an inflection point of te grap, as for f(x) = x 3 grap may oscillate in te neigborood of te point, as in te case of te function x sin(1/x), if x 0 f(x) = 0 if x = 0. at x = 0, or te Tis above grap represent te function f(x) = x 1 wit tree stationary points at x = 1, 0, 1, were ( 1, 0) and (1, 0) are local maxima and te point (0, 1) is a local minimum and at te same time a point of inflection. Te derivative of a function can, in principle, be computed from te definition by considering te difference quotient, and computing its limit. In practice, once te derivatives of a few simple functions are known, te derivatives of oter functions are more easily computed using rules for obtaining derivatives of more complicated functions from simpler ones. Most derivative computations eventually require taking te derivative of some common functions. Te following incomplete list gives some of te most frequently used functions and teir derivatives.

6 Massoud Malek Differentiation 6 Basic Derivatives y = k x r, r R y = r k x r 1. y = e c x y = c e c x. y = a c x y = c ln a c x. y = ln x, x > 0 y = 1 x y = log a x, x > 0 y = 1 x ln a y = sin x, y = cos x y = cos x, y = tan x, y = cot x, y = sec x, y = csc x, y = sin x, y = cos x, y = tan x, y = cot x, y = sec x, y = sin x y = sec x y = csc x y = sec x tan x y = csc x cot x y = cos x y = sin x y = sec x y = csc x y = sec x tan x y = csc x, y = csc x cot x y = arcsin x, y 1 = 1 x y = arccos x, y = 1 1 x y = arcsec x, y 1 = x 1 x y = arccsc x, y 1 = x 1 x y = arctan x, y = x y = arccot x, y = x Wolfram Matematica Online Differentiation website