Derivative : Differentiability

A derivative is an object that is based on, or created from, a basic or primary source. This meaning is particularly important in linguistics and etymology, where a derivative is a word that is formed from a more basic word. Similarly in chemistry a derivative is a compound that is formed from a similar compound.

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In finance, derivative is the common short form for derivative security.

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In mathematics, the derivative of a function is one of the two central concepts of calculus. The derivative of a function at a certain point is a measure of the rate at which that function is changing as an argument undergoes change. A derivative is the computation of the instantaneous slopes of f(x) at every point x. This corresponds to the slopes of the tangents to the graph of said function at said point; the slopes of such tangents can be approximated by a secant.

Functions do not have derivatives at points where they have either a vertical tangent or a discontinuity.

Differentiation and Differentiability

Differentiation can be used to determine the change which something undergoes as a result of something else changing, if a mathematical relationship betweeen two objects has been determined. The derivative of f(x) is written in several possible ways: f'(x) (pronounced f prime of x), d/dx[f(x)] (pronounced d by d x of f of x), df/dx (pronounced d f by d x), or D_x[f] (pronounced d sub x of f).

A function is differentiable at a point x if its derivative exists at this point; a function is differentiable in an interval if a derivative exists for every x within the interval. If a function isn't continuous at c, then there is no slope and the function is therefore not differentiable at c; however, even if a function is continuous at c, it may not be differentiable.

Newton's Difference Quotient

Derivatives are defined by taking the limit of a secant slope, as its two points of intersection (with f(x)) converge; the secant approaches a tangent. This is expressed by Newton's difference quotient; where h is Δx (the distance between the x-coordinates of the secant's points of intersection):

<math>f'(x)= \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}</math>

Suppose one wishes to find the derivative of a suitable function, f(x), at x. If one increases x by some small amount, Δx, one can calculate f(x + Δx). An approximation to the slope of the tangent to the curve is given by (f(x + Δx) - f(x)) / Δx, which is to say it is the change in f divided by the change in x. The smaller Δx is, the better the approximation is. From here on, Δx will be referred to as h. Mathematically, we define the derivative to be the limit[?] of this ratio, as h tends to zero.

Since immediately substituting 0, for h, results in division by zero, the numerator must be simplified such that h can be factored out and then canceled against the denominator. The resulting function, f '(x), is the derivative of f(x).

Example 1

Consider f(x) = 5:

<math>f'(x)=\lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{h} = \lim_{h\rightarrow 0} \frac{5-5}{h} = 0</math>

The derivative of a constant is zero.

Example 2

Consider the graph of <math>f(x)=2x-3</math>. Should the reader have an understanding of algebra and the Cartesian coordinate system, the reader should be able to independently determine that this line has a slope of 2 at every point. Using the above quotient (along with an understanding of the limit, secant, and tangent) one can determine the slope at (4,5):

<math>f'(4) = \lim_{h\rightarrow 0}\frac{f(4+h)-f(4)}{h} </math>

<math> = \lim_{h\rightarrow 0}\frac{2(4+h)-3-(2\cdot 4-3)}{h} </math>

<math> = \lim_{h\rightarrow 0}\frac{8+2h-3-8+3}{h} </math>

<math> = \lim_{h\rightarrow 0}\frac{2h}{h} </math>

<math> = 2 </math>

The derivative and slope are equivalent.

Example 3

Via differentiation, one can find the slope of a curve. Consider <math>f(x)=x^2</math>:

<math> f'(x) = \lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h} </math>

<math> = \lim_{h\rightarrow 0}\frac{(x+h)^2 - x^2}{h} </math>

<math> = \lim_{h\rightarrow 0}\frac{x^2 + 2xh + h^2 - x^2}{h} </math>

<math> = \lim_{h\rightarrow 0}\frac{2xh + h^2}{h} </math>

<math> = \lim_{h\rightarrow 0}(2x + h) </math>

<math> = 2x </math>

For any point x, the slope of the function <math>f(x)=x^2</math> is <math>f'(x)=2x</math>.

Example 4

Consider f(x) = √x:

<math> f'(x) = \lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h} </math>

<math> = \lim_{h\rightarrow 0}\frac{\sqrt{x+h} - \sqrt{x}}{h} </math>

<math> = \lim_{h\rightarrow 0}\frac{(\sqrt{x+h} - \sqrt{x})(\sqrt{x+h} + \sqrt{x})}{h(\sqrt{x+h} + \sqrt{x})} </math>

<math> = \lim_{h\rightarrow 0}\frac{x+h - x}{h(\sqrt{x+h} + \sqrt{x})} </math>

<math> = \lim_{h\rightarrow 0}\frac{1}{\sqrt{x+h} + \sqrt{x}} </math>

<math> = \frac{1}{2 \sqrt{x}} </math>

The Alternative Difference Quotient

Above, the derivative of f(x) (as defined by Newton) was described as the limit, as h approaches zero, of [f(x + h) - f(x)] / h. An alternative explanation of the derivative can be derived from Newton's quotient. Using the above; the derivative, at c, equals the limit, as h approaches zero, of [f(c + h) - f(c)] / h; if one then lets h = x - c (and c + h = x); then, x approaches c (as h approaches zero); thus, the derivative equals the limit, as x approaches c, of [f(x) - f(c)] / (x - c). This definition is used for a partial proof of the Chain Rule.

Stationary Points

Points on the graph of a function where the derivate equals zero are called "stationary points".

Multiple Derivatives

When the derivative of a function of x has been found, the result, being also a function of x, may be also differentiated, which gives the derivative of the derivative, or second derivative. Similarly, the derivative of the second derivative is called the third derivative, and so on. One might refer to subsequent derivatives of f by:

<math>\frac{df}{dx},\, \frac{d\left(\frac{df}{dx}\right)}{dx},\, \frac{d\left(\frac{ d\left(\frac{df}{dx}\right)}{dx}\right)}{dx}</math>

and so on.

In order to avoid such "cumbersome" notation, the following options are often preferred:

• f '(x); f ''(x); f '''(x)

or

• f⁽¹⁾(x); f⁽²⁾(x); f⁽³⁾(x)

Notable Derivatives

• For logarithmic functions:
  • The derivative of e^x, is e^x
  • The derivative of ln x, is 1 / x.
• For trigonometric functions
  • The derivative of sinx, is cosx.
  • The derivative of cosx, is -sinx.
  • The derivative of tanx, is sec²x.
  • The derivative of cotx, is csc²x.
  • The derivative of secx, is (secx)(tanx).
  • The derivative of cscx, is -(cscx)(cotx).

Algebraic Manipulation

"Messy" limit calculations can be avoided, in certain cases, because of differentiation rules which allow one to find derivatives via algebraic manipulation; rather than by direct application of Newton's difference quotient. One should not infer that the definition of derivatives, in terms of limits, is unnecessary. Rather, that definition is the means of proving the following "powerful differentiation rules"; these rules are derived from the difference quotient.

• Constant Rule: The derivative of any constant is zero.
  • Constant Multiple Rule: If c is some real number; then, the derivative of cf(x) equals c multiplied by the derivative of f(x) (see also: Product rule)
• Linearity: (a f + b g)' = a f ' + b g ' for all functions f and g and all real numbers a and b.
• General Power Rule (Polynomial rule): If f(x) = x^r, for some real number r; f '(x) = r x^r-1.
• Product Rule: (f g)' = f ' g + f g for all functions f and g.
• Quotient Rule: (f / g)' = (f ' g - f g) / g²
• Chain Rule: If f(x) = h(g(x)), then f '(x) = h '[g(x)] · g '(x)
• Inverse functions and differentiation: If y = f(x) then x = f^-1(y). Assume that f(x) and its inverse are continuous and differentiable. For cases in which Δx ≠ 0 when Δy ≠ 0, dy/dx=1/(dx/dy)
• Derivative of one variable with respect to another when both are functions of a third variable[?]: Let x = f(t) and y = g(t). Now Δy/Δx = (Δy/Δt)/(Δx/Δt) (Assume that Δx ≠ 0 when Δy ≠ 0). Hence, on letting Δt → 0, dy/dx = (dy/dt)/(dx/dt).
• Implicit differentiation: If f(x, y) = 0 be an implicit function, we have: dy/dx = - (∂f / ∂x) / (∂f / ∂y).

In addition, the derivatives of some common functions are useful to know. See the table of derivatives.

As an example, the derivative of f(x) = 2 x⁴ + sin(x²) - ln(x) e^x + 7 is f '(x) = 8 x³ + 2x cos(x²) - 1/x e^x - ln(x) e^x.

Physics

Arguably the most important application of calculus, to physics, is the concept of the "time derivative" -- the rate of change over time -- which is required for the precise definition of several important concepts. In particular, the time derivatives of an object's position are significant in Newtonian physics:

• Velocity (instantaneous velocity; the concept of average velocity predates calculus) is the derivative (with repsect to time) of an object's position.
• Acceleration is the derivative (with respect to time) of an object's velocity.
• Jerk is the derivative (with respect to time) of an object's acceleration.

Although the "time derivative" can be written "d/dt", it also has a special notation: a dot placed over the symbol of the object whose time derivative is being taken.

For example, if an object's position p(t) = -16t² + 16t + 32; then, the object's velocity is p ' (t) = -32t + 16; the object's acceleration is p '' (t) = -32; and the object's jerk is p '''(t) = 0.

If the velocity of a car is given, as a function of time; then, the derivative of said function describes the acceleration of said car, as a function of time.

Using Derivatives to Graph Functions

Derivatives are a useful tool for examining the graphs of functions. In particular, the points in the interior of the domain of a real-valued function which take that function to local extrema will all have a first derivative of zero. However, not all "critical points" (points at which the derivative of the function has determinant zero) are mapped to local extrema; some are so-called "saddle points". The Second Derivative Test is one way to evaluate critical points: if the second derivative of the function at the critical point is positive, then the point is a local minimum; if it is negative, the point is a local maximum; if it is neither, the point is either saddle point or part of a locally flat area (possibly still a local extremum, but not absolutely so). (In the case of multidimensional domains, the function will have a partial derivative of zero with respect to each dimension, at local extrema.)

Once the local extrema have been found, it is usually rather easy to get a rough idea of the general graph of the function, since (in the single-dimensional domain case) it will be uniformly increasing or decreasing except at critical points, and hence (assuming it is continuous) will have values in between its values at the critical points on either side. Also, the supremum of a continuous function on an open and bounded domain will also be one of the local maxima; the infemum will be one of the local minima--this gives one an easy way to find the bounds of the function's range.

More Info

Where a function depends on more than one variable, the concept of a partial derivative is used. Partial derivatives can be thought of informally as taking the derivative of the function with all but one variable held temporarily constant near a point. Partial derivatives are represented as ∂/∂x (where ∂ is a rounded 'd' known as the 'partial derivative symbol'). Mathematicians tend to speak the partial derivative symbol as 'der' rather than the 'dee' used for the standard derivative symbol, 'd'.

The concept of derivative can be extended to more general settings. The common thread is that the derivative at a point serves as a linear approximation of the function at that point. Perhaps the most natural situation is that of functions between differentiable manifolds; the derivative at a certain point then becomes a linear transformation between the corresponding tangent spaces and the derivative function becomes a map between the tangent bundles.

In order to differentiate all continuous functions and much more, one defines the concept of distribution.

For differentiation of complex functions of a complex variable see also Holomorphic function.

See also: differintegral.

References

• Calculus of a Single Variable: Early Transcendental Functions[?] (3rd Edition) by Edwards[?], Hostetler[?], and Larson[?] (2003)

Common misspelling and questions (FAQ)

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