DERIVATIVES : DERIVATIVES MAANAK
In calculus, a branch of mathematics, 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 one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's instantaneous velocity.: In calculus, a branch of mathematics, 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 one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's instantaneous velocity .
PowerPoint Presentation: The derivative of a function at a chosen input value describes the best linear approximation of the function near that input value. Informally, the derivative is the ratio of the infinitesimal change of the output over the infinitesimal change of the input producing that change of output. 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. In higher dimensions, the derivative of a function at a point is a linear transformation called the linearization. A closely related notion is the differential of a function.
PowerPoint Presentation: The process of finding a derivative is called differentiation . The reverse process is called antidifferentiation . The fundamental theorem of calculus states that antidifferentiation is the same as integration. Differentiation and integration constitute the two fundamental operations in single-variable calculus.
PowerPoint Presentation: THE DERIVATIVE AS A FUNCTION Let f be a function that has a derivative at every point a in the domain of f . Because every point a has a derivative, there is a function that sends the point a to the derivative of f at a . This function is written f ′( x ) and is called the derivative function or the derivative of f . The derivative of f collects all the derivatives of f at all the points in the domain of f . Sometimes f has a derivative at most, but not all, points of its domain. The function whose value at a equals f ′( a ) whenever f ′( a ) is defined and elsewhere is undefined is also called the derivative of f . It is still a function, but its domain is strictly smaller than the domain of f . Using this idea, differentiation becomes a function of functions: The derivative is an operator whose domain is the set of all functions that have derivatives at every point of their domain and whose range is a set of functions. If we denote this operator by D , then D ( f ) is the function f ′( x ). Since D ( f ) is a function, it can be evaluated at a point a . By the definition of the derivative function, D ( f )( a ) = f ′( a ).
PowerPoint Presentation: Higher derivatives Let f be a differentiable function, and let f ′( x ) be its derivative. The derivative of f ′( x ) (if it has one) is written f ′′( x ) and is called the second derivative of f . Similarly, the derivative of a second derivative, if it exists, is written f ′′′( x ) and is called the third derivative of f . These repeated derivatives are called higher-order derivatives . If x ( t ) represents the position of an object at time t , then the higher-order derivatives of x have physical interpretations. The second derivative of x is the derivative of x ′( t ), the velocity, and by definition this is the object's acceleration. The third derivative of x is defined to be the jerk, and the fourth derivative is defined to be the jounce.
PowerPoint Presentation: A function f need not have a derivative, for example, if it is not continuous. Similarly, even if f does have a derivative, it may not have a second derivative. For example, let
PowerPoint Presentation: Calculation shows that f is a differentiable function whose derivative, f′(x) is twice the absolute value function, and it does not have a derivative at zero. Similar examples show that a function can have k derivatives for any non-negative integer k but no ( k + 1) th -order derivative. A function that has k successive derivatives is called k times differentiable . If in addition the k derivative is continuous, then the function is said to be of differentiability class C k . (This is a stronger condition than having k derivatives. For an example, see differentiability class.) A function that has infinitely many derivatives is called infinitely differentiable or smooth . On the real line, every polynomial function is infinitely differentiable. By standard differentiation rules, if a polynomial of degree n is differentiated n times, then it becomes a constant function. All of its subsequent derivatives are identically zero. In particular, they exist, so polynomials are smooth functions.
PowerPoint Presentation: The derivatives of a function f at a point x provide polynomial approximations to that function near x . For example, if f is twice differentiable, then in the sense that, If f is infinitely differentiable, then this is the beginning of the Taylor series for f evaluated at x + h around x .
PowerPoint Presentation: INFLECTION POINT A point where the second derivative of a function changes sign is called an inflection point . At an inflection point, the second derivative may be zero, as in the case of the inflection point x = 0 of the function y = x 3 , or it may fail to exist, as in the case of the inflection point x = 0 of the function y = x 1/3 . At an inflection point, a function switches from being a convex function to being a concave function or vice versa.
PowerPoint Presentation: NOTATIONS FOR DIFFERENTIATION Leibniz's notation The notation for derivatives introduced by Gottfried Leibniz is one of the earliest. It is still commonly used when the equation y = f ( x ) is viewed as a functional relationship between dependent and independent variables. Then the first derivative is denoted by
PowerPoint Presentation: and was once thought of as an infinitesimal quotient. Higher derivatives are expressed using the notation, for the n th derivative of y = f ( x ) (with respect to x ). These are abbreviations for multiple applications of the derivative operator. For example,
PowerPoint Presentation: With Leibniz's notation, we can write the derivative of y at the point x = a in two different ways: Leibniz's notation allows one to specify the variable for differentiation (in the denominator). This is especially relevant for partial differentiation. It also makes the chain rule easy to remember:
PowerPoint Presentation: 2 . Lagrange's notation Sometimes referred to as prime notation , one of the most common modern notations for differentiation is due to Joseph-Louis Lagrange and uses the prime mark, so that the derivative of a function f ( x ) is denoted f ′( x ) or simply f ′. Similarly, the second and third derivatives are denoted, and
PowerPoint Presentation: 3. Newton's notation Newton's notation for differentiation, also called the dot notation, places a dot over the function name to represent a time derivative. If y = f ( t ), then, and denote, respectively, the first and second derivatives of y with respect to t . This notation is used exclusively for time derivatives, meaning that the independent variable of the function represents time. It is very common in physics and in mathematical disciplines connected with physics such as differential equations. While the notation becomes unmanageable for high-order derivatives, in practice only very few derivatives are needed.
PowerPoint Presentation: 4. Euler's notation Euler's notation uses a differential operator D , which is applied to a function f to give the first derivative Df . The second derivative is denoted D 2 f , and the n th derivative is denoted D n f . If y = f ( x ) is a dependent variable, then often the subscript x is attached to the D to clarify the independent variable x . Euler's notation is then written, or although this subscript is often omitted when the variable x is understood, for instance when this is the only variable present in the expression. Euler's notation is useful for stating and solving linear differential equations.
DERIVATIVES OF ELEMENTARY FUNCTIONS: DERIVATIVES OF ELEMENTARY FUNCTIONS Derivatives of powers : if where r is any real number, then wherever this function is defined. For example, if, then and the derivative function is defined only for positive x , not for x = 0. When r = 0, this rule implies that f ′( x ) is zero for x ≠ 0, which is almost the constant rule.
RULES FOR FINDING THE DERIVATIVE: RULES FOR FINDING THE DERIVATIVE In many cases, complicated limit calculations by direct application of Newton's difference quotient can be avoided using differentiation rules. Some of the most basic rules are the following. Constant rule : if f ( x ) is constant, then Sum rule : for all functions f and g and all real numbers α and β .
PowerPoint Presentation: Quotient rule: for all functions f and g at all inputs where g ≠ 0. Chain rule: , then
DERIVATIVES OF VECTOR VALUED FUNCTIONS: DERIVATIVES OF VECTOR VALUED FUNCTIONS A vector-valued function y ( t ) of a real variable sends real numbers to vectors in some vector space R n . A vector-valued function can be split up into its coordinate functions y 1 ( t ), y 2 ( t ), …, y n ( t ), meaning that y ( t ) = ( y 1 ( t ), ..., y n ( t )). This includes, for example, parametric curves in R 2 or R 3 . The coordinate functions are real valued functions, so the above definition of derivative applies to them. The derivative of y ( t ) is defined to be the vector, called the tangent vector, whose coordinates are the derivatives of the coordinate functions. That is,
PowerPoint Presentation: Equivalently, if the limit exists. The subtraction in the numerator is subtraction of vectors, not scalars. If the derivative of y exists for every value of t , then y ′ is another vector valued function. If e 1 , …, e n is the standard basis for R n , then y ( t ) can also be written as y 1 ( t ) e 1 + … + y n ( t ) e n . If we assume that the derivative of a vector-valued function retains the linearity property, then the derivative of y ( t ) must be, because each of the basis vectors is a constant.
PARTIAL DERIVATIVES: PARTIAL DERIVATIVES Suppose that f is a function that depends on more than one variable. For instance, f can be reinterpreted as a family of functions of one variable indexed by the other variables: In other words, every value of x chooses a function, denoted f x , which is a function of one real number. That is,
PowerPoint Presentation: Once a value of x is chosen, say a , then f ( x , y ) determines a function f a that sends y to a 2 + ay + y 2 : In this expression, a is a constant , not a variable , so f a is a function of only one real variable. Consequently the definition of the derivative for a function of one variable applies:
PowerPoint Presentation: The above procedure can be performed for any choice of a . Assembling the derivatives together into a function gives a function that describes the variation of f in the y direction: This is the partial derivative of f with respect to y . Here ∂ is a rounded d called the partial derivative symbol . To distinguish it from the letter d , ∂ is sometimes pronounced " der ", "del", or "partial" instead of " dee ".
PowerPoint Presentation: In general, the partial derivative of a function f ( x 1 , …, x n ) in the direction x i at the point ( a 1 …, a n ) is defined to be: In the above difference quotient, all the variables except x i are held fixed. That choice of fixed values determines a function of one variable, and by definition,
DIRECTIONAL DERIVATIVES: DIRECTIONAL DERIVATIVES If f is a real-valued function on R n , then the partial derivatives of f measure its variation in the direction of the coordinate axes. For example, if f is a function of x and y , then its partial derivatives measure the variation in f in the x direction and the y direction. They do not, however, directly measure the variation of f in any other direction, such as along the diagonal line y = x . These are measured using directional derivatives. Choose a vector
PowerPoint Presentation: The directional derivative of f in the direction of v at the point x is the limit, In some cases it may be easier to compute or estimate the directional derivative after changing the length of the vector. Often this is done to turn the problem into the computation of a directional derivative in the direction of a unit vector. To see how this works, suppose that v = λ u . Substitute h = k /λ into the difference quotient. The difference quotient becomes:
PowerPoint Presentation: This is λ times the difference quotient for the directional derivative of f with respect to u . Furthermore, taking the limit as h tends to zero is the same as taking the limit as k tends to zero because h and k are multiples of each other. Therefore D v ( f ) = λ D u ( f ). Because of this rescaling property, directional derivatives are frequently considered only for unit vectors. If all the partial derivatives of f exist and are continuous at x , then they determine the directional derivative of f in the direction v by the formula:
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