Differential calculus

The graph of a function, drawn in black, and a tangent line to that function, drawn in red. The slope of the tangent line equals the derivative of the function at the marked point.

Inmathematics,differential calculusis a subfield ofcalculusthat studies the rates at which quantities change.It is one of the two traditional divisions of calculus, the other beingintegral calculus—the study of the area beneath a curve.

The primary objects of study in differential calculus are thederivativeof afunction,related notions such as thedifferential,and their applications. The derivative of a function at a chosen input value describes therate of changeof the function near that input value. The process of finding a derivative is calleddifferentiation.Geometrically, the derivative at a point is theslopeof thetangent lineto thegraph of the functionat that point, provided that the derivative exists and is defined at that point. For areal-valued functionof a single real variable, the derivative of a function at a point generally determines the bestlinear approximationto the function at that point.

Differential calculus andintegral calculusare connected by thefundamental theorem of calculus.This states that differentiation is the reverse process tointegration.

Differentiation has applications in nearly all quantitative disciplines. Inphysics,the derivative of thedisplacementof a moving body with respect to time is thevelocityof the body, and the derivative of the velocity with respect to time isacceleration.The derivative of themomentumof a body with respect totimeequals the force applied to the body; rearranging this derivative statement leads to the famousF = maequation associated withNewton's second law of motion.Thereaction rateof achemical reactionis a derivative. Inoperations research,derivatives determine the most efficient ways to transport materials and design factories.

Derivatives are frequently used to find themaxima and minimaof a function. Equations involving derivatives are calleddifferential equationsand are fundamental in describingnatural phenomena.Derivatives and their generalizations appear in many fields of mathematics, such ascomplex analysis,functional analysis,differential geometry,measure theory,andabstract algebra.

Derivative

Main article:Derivative

The graph of an arbitrary function$y=f(x)$.The orange line is tangent to$x=a$,meaning at that exact point, the slope of the curve and the straight line are the same.

The derivative at different points of a differentiable function

The derivative of$f(x)$at the point$x=a$is theslopeof the tangent to$(a,f(a))$.In order to gain an intuition for this, one must first be familiar with finding the slope of a linear equation, written in the form$y=mx+b$.The slope of an equation is its steepness. It can be found by picking any two points and dividing the change in$y$by the change in$x$,meaning that${\text{slope }}={\frac {{\text{ change in }}y}{{\text{change in }}x}}$.For, the graph of$y=-2x+13$has a slope of$-2$,as shown in the diagram below:

The graph of$y=-2x+13$

${\frac {{\text{change in }}y}{{\text{change in }}x}}={\frac {-6}{+3}}=-2$

For brevity,${\frac {{\text{change in }}y}{{\text{change in }}x}}$is often written as${\frac {\Delta y}{\Delta x}}$,with$\Delta$being the Greek letter delta, meaning 'change in'. The slope of a linear equation is constant, meaning that the steepness is the same everywhere. However, many graphs such as$y=x^{2}$vary in their steepness. This means that you can no longer pick any two arbitrary points and compute the slope. Instead, the slope of the graph can be computed by considering the tangent line—a line that 'just touches' a particular point.The slope of a curve at a particular point is equal to the slope of the tangent to that point. For example,$y=x^{2}$has a slope of$4$at$x=2$because the slope of the tangent line to that point is equal to$4$:

The graph of$y=x^{2}$,with a straight line that is tangent to$(2,4)$.The slope of the tangent line is equal to$4$.(The axes of the graph do not use a 1:1 scale.)

The derivative of afunctionis then simply the slope of this tangent line.Even though the tangent line only touches a single point at the point of tangency, it can be approximated by a line that goes through two points. This is known as asecant line.If the two points that the secant line goes through are close together, then the secant line closely resembles the tangent line, and, as a result, its slope is also very similar:

The dotted line goes through the points$(2,4)$and$(3,9)$,which both lie on the curve$y=x^{2}$.Because these two points are fairly close together, the dotted line and tangent line have a similar slope. As the two points become closer together, the error produced by the secant line becomes vanishingly small.

The advantage of using a secant line is that its slope can be calculated directly. Consider the two points on the graph$(x,f(x))$and$(x+\Delta x,f(x+\Delta x))$,where$\Delta x$is a small number. As before, the slope of the line passing through these two points can be calculated with the formula${\text{slope }}={\frac {\Delta y}{\Delta x}}$.This gives

${\text{slope}}={\frac {f(x+\Delta x)-f(x)}{\Delta x}}$

As$\Delta x$gets closer and closer to$0$,the slope of the secant line gets closer and closer to the slope of the tangent line. This is formally written as

$\lim _{\Delta x\to 0}{\frac {f(x+\Delta x)-f(x)}{\Delta x}}$

The above expression means 'as$\Delta x$gets closer and closer to 0, the slope of the secant line gets closer and closer to a certain value'. The value that is being approached is the derivative of$f(x)$;this can be written as$f'(x)$.If$y=f(x)$,the derivative can also be written as${\frac {dy}{dx}}$,with$d$representing aninfinitesimalchange. For example,$dx$represents an infinitesimal change in x.In summary, if$y=f(x)$,then the derivative of$f(x)$is

${\frac {dy}{dx}}=f'(x)=\lim _{\Delta x\to 0}{\frac {f(x+\Delta x)-f(x)}{\Delta x}}$

provided such a limit exists.We have thus succeeded in properly defining the derivative of a function, meaning that the 'slope of the tangent line' now has a precise mathematical meaning. Differentiating a function using the above definition is known as differentiation from first principles. Here is a proof, using differentiation from first principles, that the derivative of$y=x^{2}$is$2x$:

${\begin{aligned}{\frac {dy}{dx}}&=\lim _{\Delta x\to 0}{\frac {f(x+\Delta x)-f(x)}{\Delta x}}\\&=\lim _{\Delta x\to 0}{\frac {(x+\Delta x)^{2}-x^{2}}{\Delta x}}\\&=\lim _{\Delta x\to 0}{\frac {x^{2}+2x\Delta x+(\Delta x)^{2}-x^{2}}{\Delta x}}\\&=\lim _{\Delta x\to 0}{\frac {2x\Delta x+(\Delta x)^{2}}{\Delta x}}\\&=\lim _{\Delta x\to 0}2x+\Delta x\\\end{aligned}}$

As$\Delta x$approaches$0$,$2x+\Delta x$approaches$2x$.Therefore,${\frac {dy}{dx}}=2x$.This proof can be generalised to show that${\frac {d(ax^{n})}{dx}}=anx^{n-1}$if$a$and$n$areconstants.This is known as thepower rule.For example,${\frac {d}{dx}}(5x^{4})=5(4)x^{3}=20x^{3}$.However, many other functions cannot be differentiated as easily aspolynomial functions,meaning that sometimes further techniques are needed to find the derivative of a function. These techniques include thechain rule,product rule,andquotient rule.Other functions cannot be differentiated at all, giving rise to the concept ofdifferentiability.

A closely related concept to the derivative of a function is itsdifferential.Whenxandyare real variables, the derivative offatxis the slope of the tangent line to the graph offatx.Because the source and target offare one-dimensional, the derivative offis a real number. Ifxandyare vectors, then the best linear approximation to the graph offdepends on howfchanges in several directions at once. Taking the best linear approximation in a single direction determines apartial derivative,which is usually denoted∂y/∂x.The linearization offin all directions at once is called thetotal derivative.

History of differentiation

Main article:History of calculus

The concept of a derivative in the sense of atangent lineis a very old one, familiar to ancientGreekmathematicians such asEuclid(c. 300 BC),Archimedes(c. 287–212 BC), andApollonius of Perga(c. 262–190 BC).Archimedesalso made use ofindivisibles,although these were primarily used to study areas and volumes rather than derivatives and tangents (seeThe Method of Mechanical Theorems). The use of infinitesimals to compute rates of change was developed significantly byBhāskara II(1114–1185); indeed, it has been arguedthat many of the key notions of differential calculus can be found in his work, such as "Rolle's theorem".

The mathematician,Sharaf al-Dīn al-Tūsī(1135–1213), in hisTreatise on Equations,established conditions for some cubic equations to have solutions, by finding the maxima of appropriate cubic polynomials. He obtained, for example, that the maximum (for positivex) of the cubicax²–x³occurs whenx= 2a/ 3,and concluded therefrom that the equationax²=x³+ chas exactly one positive solution whenc= 4a³/ 27,and two positive solutions whenever0 <c< 4a³/ 27.The historian of science,Roshdi Rashed,has argued that al-Tūsī must have used the derivative of the cubic to obtain this result. Rashed's conclusion has been contested by other scholars, however, who argue that he could have obtained the result by other methods which do not require the derivative of the function to be known.

The modern development of calculus is usually credited toIsaac Newton(1643–1727) andGottfried Wilhelm Leibniz(1646–1716), who provided independentand unified approaches to differentiation and derivatives. The key insight, however, that earned them this credit, was thefundamental theorem of calculusrelating differentiation and integration: this rendered obsolete most previous methods for computing areas and volumes.For their ideas on derivatives, both Newton and Leibniz built on significant earlier work by mathematicians such asPierre de Fermat(1607-1665),Isaac Barrow(1630–1677),René Descartes(1596–1650),Christiaan Huygens(1629–1695),Blaise Pascal(1623–1662) andJohn Wallis(1616–1703). Regarding Fermat's influence, Newton once wrote in a letter that "I had the hint of this method [of fluxions] from Fermat's way of drawing tangents, and by applying it to abstract equations, directly and invertedly, I made it general."Isaac Barrow is generally given credit for the early development of the derivative.Nevertheless, Newton and Leibniz remain key figures in the history of differentiation, not least because Newton was the first to apply differentiation totheoretical physics,while Leibniz systematically developed much of the notation still used today.

Since the 17th century many mathematicians have contributed to the theory of differentiation. In the 19th century, calculus was put on a much more rigorous footing by mathematicians such asAugustin Louis Cauchy(1789–1857),Bernhard Riemann(1826–1866), andKarl Weierstrass(1815–1897). It was also during this period that the differentiation was generalized toEuclidean spaceand thecomplex plane.

The 20th century brought two major steps towards our present understanding and practice of derivation:Lebesgue integration,besides extending integral calculus to many more functions, clarified the relation between derivation and integration with the notion ofabsolute continuity.Later thetheory of distributions(afterLaurent Schwartz) extended derivation to generalized functions (e.g., theDirac delta functionpreviously introduced inQuantum Mechanics) and became fundamental to nowadays applied analysis especially by the use ofweak solutionstopartial differential equations.

Applications of derivatives

Optimization

Iffis adifferentiable functiononℝ(or anopen interval) andxis alocal maximumor alocal minimumoff,then the derivative offatxis zero. Points wheref'(x) = 0are calledcritical pointsorstationary points(and the value offatxis called acritical value). Iffis not assumed to be everywhere differentiable, then points at which it fails to be differentiable are also designated critical points.

Iffis twice differentiable, then conversely, a critical pointxoffcan be analysed by considering thesecond derivativeoffatx:

• if it is positive,xis a local minimum;
• if it is negative,xis a local maximum;
• if it is zero, thenxcould be a local minimum, a local maximum, or neither. (For example,f(x) =x³has a critical point atx= 0,but it has neither a maximum nor a minimum there, whereasf(x) = ±x⁴has a critical point atx= 0and a minimum and a maximum, respectively, there.)

This is called thesecond derivative test.An alternative approach, called thefirst derivative test,involves considering the sign of thef'on each side of the critical point.

Taking derivatives and solving for critical points is therefore often a simple way to find local minima or maxima, which can be useful inoptimization.By theextreme value theorem,a continuous function on aclosed intervalmust attain its minimum and maximum values at least once. If the function is differentiable, the minima and maxima can only occur at critical points or endpoints.

This also has applications in graph sketching: once the local minima and maxima of a differentiable function have been found, a rough plot of the graph can be obtained from the observation that it will be either increasing or decreasing between critical points.

Inhigher dimensions,a critical point of ascalar valuedfunction is a point at which thegradientis zero. Thesecond derivativetest can still be used to analyse critical points by considering theeigenvaluesof theHessian matrixof second partial derivatives of the function at the critical point. If all of the eigenvalues are positive, then the point is a local minimum; if all are negative, it is a local maximum. If there are some positive and some negative eigenvalues, then the critical point is called a "saddle point",and if none of these cases hold (i.e., some of the eigenvalues are zero) then the test is considered to be inconclusive.

Calculus of variations

Main article:Calculus of variations

One example of an optimization problem is: Find the shortest curve between two points on a surface, assuming that the curve must also lie on the surface. If the surface is a plane, then the shortest curve is a line. But if the surface is, for example, egg-shaped, then theshortest pathis not immediately clear. These paths are calledgeodesics,and one of the most fundamental problems in the calculus of variations is finding geodesics. Another example is: Find the smallest area surface filling in a closed curve in space. This surface is called aminimal surfaceand it, too, can be found using the calculus of variations.

Physics

Calculus is of vital importance in physics: many physical processes are described by equations involving derivatives, calleddifferential equations.Physics is particularly concerned with the way quantities change and develop over time, and the concept of the "time derivative"— the rate of change over time — is essential for the precise definition of several important concepts. In particular, the time derivatives of an object's position are significant inNewtonian physics:

• velocityis the derivative (with respect to time) of an object's displacement (distance from the original position)
• accelerationis the derivative (with respect to time) of an object's velocity, that is, the second derivative (with respect to time) of an object's position.

For example, if an object's position on a line is given by

$x(t)=-16t^{2}+16t+32,\,\!$

then the object's velocity is

${\dot {x}}(t)=x'(t)=-32t+16,\,\!$

and the object's acceleration is

${\ddot {x}}(t)=x''(t)=-32,\,\!$

which is constant.

Differential equations

Main article:Differential equation

A differential equation is a relation between a collection of functions and their derivatives. Anordinary differential equationis a differential equation that relates functions of one variable to their derivatives with respect to that variable. Apartial differential equationis a differential equation that relates functions of more than one variable to theirpartial derivatives.Differential equations arise naturally in the physical sciences, in mathematical modelling, and within mathematics itself. For example,Newton's second law,which describes the relationship between acceleration and force, can be stated as the ordinary differential equation

$F(t)=m{\frac {d^{2}x}{dt^{2}}}.$

Theheat equationin one space variable, which describes how heat diffuses through a straight rod, is the partial differential equation

${\frac {\partial u}{\partial t}}=\alpha {\frac {\partial ^{2}u}{\partial x^{2}}}.$

Hereu(x,t)is the temperature of the rod at positionxand timetandαis a constant that depends on how fast heat diffuses through the rod.

Mean value theorem

Main article:Mean value theorem

The mean value theorem: For each differentiable function$f:[a,b]\to \mathbb {R}$with$a<b$there is a$c\in (a,b)$with$f'(c)={\tfrac {f(b)-f(a)}{b-a}}$.

The mean value theorem gives a relationship between values of the derivative and values of the original function. Iff(x)is a real-valued function andaandbare numbers witha<b,then the mean value theorem says that under mild hypotheses, the slope between the two points(a,f(a))and(b,f(b))is equal to the slope of the tangent line tofat some pointcbetweenaandb.In other words,

$f'(c)={\frac {f(b)-f(a)}{b-a}}.$

In practice, what the mean value theorem does is control a function in terms of its derivative. For instance, suppose thatfhas derivative equal to zero at each point. This means that its tangent line is horizontal at every point, so the function should also be horizontal. The mean value theorem proves that this must be true: The slope between any two points on the graph offmust equal the slope of one of the tangent lines off.All of those slopes are zero, so any line from one point on the graph to another point will also have slope zero. But that says that the function does not move up or down, so it must be a horizontal line. More complicated conditions on the derivative lead to less precise but still highly useful information about the original function.

Taylor polynomials and Taylor series

Main articles:Taylor polynomialandTaylor series

The derivative gives the best possible linear approximation of a function at a given point, but this can be very different from the original function. One way of improving the approximation is to take a quadratic approximation. That is to say, the linearization of a real-valued functionf(x)at the pointx₀is a linearpolynomiala+b(x−x₀),and it may be possible to get a better approximation by considering a quadratic polynomiala+b(x−x₀) +c(x−x₀)².Still better might be a cubic polynomiala+b(x−x₀) +c(x−x₀)²+d(x−x₀)³,and this idea can be extended to arbitrarily high degree polynomials. For each one of these polynomials, there should be a best possible choice of coefficientsa,b,c,anddthat makes the approximation as good as possible.

In theneighbourhoodofx₀,forathe best possible choice is alwaysf(x₀),and forbthe best possible choice is alwaysf'(x₀).Forc,d,and higher-degree coefficients, these coefficients are determined by higher derivatives off.cshould always bef''(x₀)/2,anddshould always bef'''(x₀)/3!.Using these coefficients gives theTaylor polynomialoff.The Taylor polynomial of degreedis the polynomial of degreedwhich best approximatesf,and its coefficients can be found by a generalization of the above formulas.Taylor's theoremgives a precise bound on how good the approximation is. Iffis a polynomial of degree less than or equal tod,then the Taylor polynomial of degreedequalsf.

The limit of the Taylor polynomials is an infinite series called theTaylor series.The Taylor series is frequently a very good approximation to the original function. Functions which are equal to their Taylor series are calledanalytic functions.It is impossible for functions with discontinuities or sharp corners to be analytic; moreover, there existsmooth functionswhich are also not analytic.

Implicit function theorem

Main article:Implicit function theorem

Some natural geometric shapes, such ascircles,cannot be drawn as thegraph of a function.For instance, iff(x,y) =x²+y²− 1,then the circle is the set of all pairs(x,y)such thatf(x,y) = 0.This set is called the zero set off,and is not the same as the graph off,which is aparaboloid.The implicit function theorem converts relations such asf(x,y) = 0into functions. It states that iffiscontinuously differentiable,then around most points, the zero set offlooks like graphs of functions pasted together. The points where this is not true are determined by a condition on the derivative off.The circle, for instance, can be pasted together from the graphs of the two functions±√1 -x².In a neighborhood of every point on the circle except(−1, 0)and(1, 0),one of these two functions has a graph that looks like the circle. (These two functions also happen to meet(−1, 0)and(1, 0),but this is not guaranteed by the implicit function theorem.)

The implicit function theorem is closely related to theinverse function theorem,which states when a function looks like graphs ofinvertible functionspasted together.

See also

• Differential (calculus)
• Numerical differentiation
• Techniques for differentiation
• List of calculus topics
• Notation for differentiation
• Mathematics portal

References

Works cited

• Berggren, J. L. (1990). "Innovation and Tradition in Sharaf al-Din al-Tusi's Muadalat".Journal of the American Oriental Society.110(2): 304–309.doi:10.2307/604533.JSTOR604533.

Other sources

• J. Edwards (1892).Differential Calculus.London: MacMillan and Co. p.1.
• Boman, Eugene, and Robert Rogers.Differential Calculus: From Practice to Theory.2022, personal.psu.edu/ecb5/DiffCalc.pdf[1]Archived2022-12-20 at theWayback Machine.