Derivative

In Calculus , a branch of Mathematics , the derivative is a measurement of how a Function changes when the values of its inputs change. The derivative of a function at a chosen input value describes the best Linear Approximation of the function near that input value. 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 .Differential calculus, as discussed in this article, is a very well-established mathematical discipline for which there are many sources. Almost all of the material in this article can be found in Apostol 1967, Apostol 1969, and Spivak 1994.

The process of finding a derivative is called differentiation. The Fundamental Theorem Of Calculus states that differentiation is the reverse process to Integration .

DIFFERENTIATION AND THE DERIVATIVE

Differentiation is a method to compute the rate at which a quantity, ''y'', changes with respect to the change in another quantity, ''x'', upon which it is Dependent . This rate of change is called the '''derivative''' of ''y'' with respect to ''x''. In more precise language, the dependency of ''y'' on ''x'' means that ''y'' is a Function of ''x''. If ''x'' and ''y'' are Real Number s, and if the Graph of ''y'' is plotted against ''x'', the derivative measures the Slope of this graph at each point. This functional relationship is often denoted ''y'' = ''f''(''x''), where ''f'' denotes the function.

The simplest case is when ''y'' is a Linear Function of ''x'', meaning that the graph of ''y'' against ''x'' is a straight line. In this case, ''y'' = ''f''(''x'') = ''m'' ''x'' + ''c'', for real numbers ''m'' and ''c'', and the slope ''m'' is given by
:

m={\mbox{change in } y \over \mbox{change in } x} = {\Delta y \over{\Delta x}}

where the symbol Δ (the uppercase form of the Greek letter Delta ) is an abbreviation for "change in". This formula is true because

y

It follows that Δ''y'' = ''m'' Δ''x''.

This gives an exact value for the slope of a straight line. If the function ''f'' is not a straight line, however, then the change in ''y'' divided by the change in ''x'' varies: differentiation is a method to find an exact value for this rate of change at any given value of ''x''.

line at (''x'', ''f''(''x''))]]

to curve ''y''= ''f''(''x'') determined by points (''x'', ''f''(''x'')) and (''x''+''h'', ''f''(''x''+''h'')).]]

The idea, illustrated by Figures 1-3, is to compute the rate of change as the Limiting Value of the Ratio Of The Differences Δ''y'' / Δ''x'' as Δ''x'' becomes infinitely small.

In Leibniz's Notation , such an Infinitesimal change in ''x'' is denoted by ''dx'', and the derivative of ''y'' with respect to ''x'' is written
:

rac{dy}{dx} \,\!

suggesting the ratio of two infinitesimal quantities. (The above expression is pronounced in various ways such as "d y by d x" or "d y over d x". The oral form "d y d x" is often used conversationally, although it may lead to confusion.)

The most common approachSpivak 1994, chapter 10. to turn this intuitive idea into a precise definition uses Limits , but there are other methods, such as Non-standard Analysis See Differential (infinitesimal) for an overview. Further approaches include the Radon-Nikodym Theorem , and the universal derivation (see Kähler Differential )..

Definition via difference quotients

Let ''y''=''f''(''x'') be a function of ''x''. In classical geometry, the tangent line at a real number ''a'' was the unique line through the point (''a'', ''f''(''a'')) which did ''not'' meet the graph of ''f'' Transversally , meaning that the line did not pass straight through the graph. The derivative of ''y'' with respect to ''x'' at ''a'' is, geometrically, the slope of the tangent line to the graph of ''f'' at ''a''. The slope of the tangent line is very close to the slope of the line through (''a'', ''f''(''a'')) and a nearby point on the graph, for example (''a'' + ''h'', ''f''(''a'' + ''h'')). These lines are called Secant Line s. A value of ''h'' close to zero will give a good approximation to the slope of the tangent line, and smaller values (in Absolute Value ) of ''h'' will, in general, give better Approximation s. The slope of the secant line is the difference between the ''y'' values of these points divided by the difference between the ''x'' values, that is,
:

rac{f(a+h)-f(a)}{h}.

This expression is Newton 's Difference Quotient . The derivative is the value of the difference quotient as the secant lines get closer and closer to the tangent line. Formally, the '''derivative''' of the function ''f'' at ''a'' is the Limit
:

f'(a)=\lim_{h	o 0}{f(a+h)-f(a)\over h}

of the difference quotient as ''h'' approaches zero, if this limit exists. If the limit exists, then ''f'' is differentiable at ''a''. Here ''f'' '(''a'') is one of several common notations for the derivative ( See Below ).

Equivalently, the derivative satisfies the property that
:

\lim_{h	o 0}{f(a+h)-f(a) - f'(a)\cdot h\over h} = 0,

which has the intuitive interpretation (see Figure 1) that the tangent line to ''f'' at ''a'' gives the ''best Linear approximation''
:

f(a+h) \approx f(a) + f'(a)h

to ''f'' near ''a'' (i.e., for small ''h''). This interpretation is the most easy to generalize to other settings ( See Below ).

Substituting 0 for ''h'' in the difference quotient causes Division By Zero , so the slope of the tangent line cannot be found directly. Instead, define ''Q''(''h'') to be the difference quotient as a function of ''h'':
:

Q(h) = rac{f(a + h) - f(a)}{h}

.
''Q''(''h'') is the slope of the secant line between (''a'', ''f''(''a'')) and (''a'' + ''h'', ''f''(''a'' + ''h'')). If ''f'' is a Continuous Function , meaning that its graph is an unbroken curve with no gaps, then ''Q'' is a continuous function away from the point ''h'' = 0. If the limit

extstyle\lim_{h	o 0} Q(h)

exists, meaning that there is a way of choosing a value for ''Q''(0) which makes the graph of ''Q'' a continuous function, then the function ''f'' is differentiable at the point ''a'', and its derivative at ''a'' equals ''Q''(0).

In practice, the continuity of the difference quotient ''Q''(''h'') at ''h'' = 0 is shown by modifying the numerator to cancel ''h'' in the denominator. This process can be long and tedious for complicated functions, and many Short Cuts are commonly used to simplify the process.

Example

The squaring function ''f''(''x'') = ''x''² is differentiable at ''x'' = 3, and its derivative there is 6. This is proven by writing the difference quotient as follows:

:

{f(3+h)-f(3)\over h} = {(3+h)^2 - 9\over{h}} = {9 + 6h + h^2 - 9\over{h}} = {6h + h^2\over{h}} = 6 + h.

From the last expression, we see that the difference quotient equals ''6'' + ''h'' when ''h'' is not zero and is undefined when ''h'' is zero. (Remember that in the definition of the difference quotient, we divided by ''h'', so the difference quotient is always undefined when ''h'' is zero.) However, there is a natural way of filling in a value for the difference quotient at zero, namely ''6''. Hence the slope of the graph of the squaring function at the point (3, 9) is 6, and so its derivative at ''x'' = 3 is ''f'' '(3) = 6.

More generally, a similar computation shows that the derivative of the squaring function at ''x'' = ''a'' is ''f'' '(''a'') = 2''a''.

Continuity and differentiability

If ''y'' = ''f''(''x'') is differentiable at ''a'', then ''f'' must also be Continuous at ''a''. As an example, choose a point ''a'' and let ''f'' be the Step Function which returns a value, say 1, for all ''x'' less than ''a'', and returns a different value, say 10, for all ''x'' greater than or equal to ''a''. ''f'' cannot have a derivative at ''a''. If ''h'' is negative, then ''a'' + ''h'' is on the low part of the step, so the secant line from ''a'' to ''a'' + ''h'' will be very steep, and as ''h'' tends to zero the slope tends to infinity. If ''h'' is positive, then ''a'' + ''h'' is on the high part of the step, so the secant line from ''a'' to ''a'' + ''h'' will have slope zero. Consequently the secant lines do not approach any single slope, so the limit of the difference quotient does not exist.Despite this, it is still possible to take the derivative in the sense of Distributions . The result is nine times the Dirac Measure centered at ''a''.

In other words, the different choices of ''a'' index a family of one-variable functions just as in the example above. This expression also shows that the computation of partial derivatives reduces to the computation of one-variable derivatives.

An important example of a function of several variables is the case of a Scalar-valued Function ''f''(''x''₁,...''x''_''n'') on a domain in Euclidean space R^''n'' (e.g., on R² or R³). In this case ''f'' has a partial derivative ∂''f''/∂''x''_''j'' with respect to each variable ''x''_''j''. At the point ''a'', these partial derivatives define the vector
:

abla f(a) = \left(rac{\partial f}{\partial x_1}(a), \ldots, rac{\partial f}{\partial x_n}(a)
ight).

This vector is called the Gradient of ''f'' at ''a''. If ''f'' is differentiable at every point in some domain, then the gradient is a vector-valued function ∇''f'' which takes the point ''a'' to the vector ∇''f(a)''. Consequently the gradient determines a Vector Field .

Directional derivatives

See Also: Directional derivative

If ''f'' is a real-valued function on Rⁿ, then the partial derivatives of ''f'' measure its variation in the direction of the coordinate axes. For instance, 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
:

\mathbf{v} = (v_1,\ldots,v_n).

The directional derivative of ''f'' in the direction of '''v''' at the point '''x''' is the limit
:

D_{\mathbf{v}}{f}(\boldsymbol{x}) = \lim_{h 
ightarrow 0}{rac{f(\boldsymbol{x} + h\mathbf{v}) - f(\boldsymbol{x})}{h}}.

Let λ be a scalar. The substitution of ''h/λ'' for ''h'' changes the λv direction's difference quotient into λ times the v direction's difference quotient. Consequently, the directional derivative in the λv direction is λ times the directional derivative in the v direction. Because of this, directional derivatives are often considered only for unit vectors v.

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:
:

D_{\mathbf{v}}{f}(\boldsymbol{x}) = \sum_{j=1}^n v_j rac{\partial f}{\partial x_j}.

This is a consequence of the definition of the Total Derivative . It follows that the directional derivative is Linear in v.

The same definition also works when ''f'' is a function with values in R^m. In this case, the directional derivative is a vector in R^m.

The total derivative, the Jacobian, and the differential

See Also: Total derivative

v.) In other words, if ''v'' is measured in terms of distances on the target, then, because ''v'' can only be measured through ''f'', ''v'' no longer appears to be a unit vector because ''f'' does not preserve unit vectors. Instead ''v'' appears to have length ''f'(a)''. If ''m'' is greater than one, then by writing ''f'' using coordinate functions, the length of ''v'' in each of the coordinate directions can be measured separately.

Suppose now that ''f'' is a function from a domain in R^''n'' to R^''m'' and that '''a''' is a point in the domain of ''f''. The derivative of ''f'' at '''a''' should still be the best linear approximation to ''f'' at '''a'''. In other words, if ''v'' is a vector on R^''n'', then ''f'''('''a''') should be the Linear Transformation that best approximates ''f''. The linear transformation should contain all the information about how ''f'' transforms vectors at '''a''' to vectors at ''f('''a''')'', and in symbols, this means it should be the linear transformation ''f'('''a''')'' such that

The existence of the Jacobian is strictly stronger than existence of all the partial derivatives, but if the partial derivatives exist and satisfy mild smoothness conditions, then the total derivative exists and is given by the Jacobian.

The transpose of the Jacobian matrix determines a linear map from R^''m'' to R^''n''. More intrinsically, this is the dual map on Dual Vector Spaces . This linear map is called the '''differential''' of ''f'' at '''a'''.

The definition of the total derivative subsumes the definition of the derivative in one variable. In this case, the total derivative exists if and only if the usual derivative exists. The Jacobian matrix reduces to a 1×1 matrix whose only entry is the derivative ''f'(x)''. This 1×1 matrix satisfies the property that ''f''(''a'' + ''h'') − ''f''(''a'') − ''f'' '(''a'')''h'' is approximately zero, in other words that

:

f(a+h) \approx f(a) + f'(a)h.

Up to changing variables, this is the statement that the function

x \mapsto f(a) + f'(a)(x-a)

is the best linear approximation to ''f'' at ''a''.

The total derivative of a function does not give another function in the same way that one-variable case. This is because the total derivative of a multivariable function has to record much more information than the derivative of a single-variable function. Instead, the total derivative gives a function from the Tangent Bundle of the source to the tangent bundle of the target.

GENERALIZATIONS

See Also: Derivative (generalizations)

The concept of a derivative can be extended to many other settings. The common thread is that the derivative of a function at a point serves as a Linear Approximation of the function at that point.

An important generalization of the derivative concerns — see Holomorphic Function s.

Another generalization concerns functions between from the tangent space of ''M'' at ''x'' to the tangent space of ''N'' at ''f''(''x''). The derivative function becomes a map between the Tangent Bundle s of ''M'' and ''N''. This definition is fundamental in Differential Geometry and has many uses — see Pushforward (differential) and Pullback (differential Geometry) .

Differentiation can also be defined for maps between Infinite Dimensional Vector Space s such as Banach Space s and Fréchet Space s. There is a generalization both of the directional derivative, called the Gâteaux Derivative , and of the differential, called the Fréchet Derivative .

One deficiency of the classical derivative is that not very many functions are differentiable. Nevertheless, there is a way of extending the notion of the derivative so that all Continuous functions and many other functions can be differentiated using a concept known as the Weak Derivative . The idea is to embed the continuous functions in a larger space called the space of Distributions and only require that a function is differentiable "on average".

The properties of the derivative have inspired the introduction and study of many similar objects in algebra and topology — see, for example, Differential Algebra .

NOTES

REFERENCES

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	Last	Anton
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Online books

	Last	Crowell
	First	Benjamin
	Title	Calculus
	Date	2003
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	Last	Garrett
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	Date	2004
	Title	Notes on First-Year Calculus
	Url	http://wwwmathumnedu/~garrett/calculus/

	Last	Hussain
	First	Faraz
	Date	2006
	Title	Understanding Calculus
	Url	http://wwwunderstandingcalculuscom/

	Last	Keisler
	First	H Jerome
	Date	2000
	Title	Elementary Calculus: An Approach Using Infinitesimals
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	Last	Mauch
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	Title	Unabridged Version of Sean's Applied Math Book
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	Last	Sloughter
	First	Dan
	Date	2000
	Title	Difference Equations to Differential Equations
	Url	http://mathfurmanedu/~dcs/book/

	Last	Stroyan
	First	Keith D
	Date	1997
	Title	A Brief Introduction to Infinitesimal Calculus
	Url	http://wwwmathuiowaedu/~stroyan/InfsmlCalculus/InfsmlCalchtm

	Last	Wikibooks
	Title	Calculus
	Url	http://enwikibooksorg/wiki/Calculus

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