EXAMPLE 10

Read QR Code 2d Barcode In NoneUsing Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications.

Encoding QR Code In NoneUsing Barcode printer for Software Control to generate, create QR Code ISO/IEC18004 image in Software applications.

f[x_] = x5 + x4 + x3 + x2 + x + 1; Derivative[1][f] 1 + 2#1 + 3#1 2 + 4#1 3 + 5#1 4 & Derivative[1][f][x] 1 + 2x + 3x 2 + 4x 3 + 5x 4

QR Code Recognizer In NoneUsing Barcode decoder for Software Control to read, scan read, scan image in Software applications.

QR Code Creator In Visual C#.NETUsing Barcode drawer for Visual Studio .NET Control to generate, create QR Code image in .NET framework applications.

Mathematica returns a pure function representing the derivative of f. Pure functions are discussed in the appendix. #1 is replaced by x.

QR Code ISO/IEC18004 Generator In VS .NETUsing Barcode creation for ASP.NET Control to generate, create Quick Response Code image in ASP.NET applications.

Make QR Code 2d Barcode In .NET FrameworkUsing Barcode creation for .NET framework Control to generate, create QR Code 2d barcode image in .NET framework applications.

The numerical value of a derivative at a specific point can be computed several different ways, depending upon how the derivative is computed. The next example illustrates the most common techniques.

Draw QR Code 2d Barcode In Visual Basic .NETUsing Barcode printer for .NET framework Control to generate, create Quick Response Code image in .NET framework applications.

Encode Code39 In NoneUsing Barcode maker for Software Control to generate, create Code 3/9 image in Software applications.

EXAMPLE 11

Print DataMatrix In NoneUsing Barcode drawer for Software Control to generate, create Data Matrix ECC200 image in Software applications.

GTIN - 13 Creation In NoneUsing Barcode generation for Software Control to generate, create EAN13 image in Software applications.

f[x_]= (x2 x + 1)5; f''[1] 30 D[f[x] {x, 2}] /. x 1 , 30 {x, 2}f[x] /. x 1 30 g Derivative[2][f] g[1] 30 f[x_] = x3 g[1] 6

Paint UCC.EAN - 128 In NoneUsing Barcode encoder for Software Control to generate, create GS1-128 image in Software applications.

Making Bar Code In NoneUsing Barcode generator for Software Control to generate, create bar code image in Software applications.

Here we have de ned a new function, g, as the second derivative of f. If f is changed, g will be the second derivative of the new function. Note the use of here. This is crucial if g is to re ect the change in f. In each of the rst three parts of this example, the second derivative is computed and then x is replaced by 1.

ISSN Generator In NoneUsing Barcode creation for Software Control to generate, create ISSN image in Software applications.

DataMatrix Recognizer In JavaUsing Barcode recognizer for Java Control to read, scan read, scan image in Java applications.

Differential Calculus

Barcode Recognizer In Visual Basic .NETUsing Barcode Control SDK for .NET Control to generate, create, read, scan barcode image in VS .NET applications.

Bar Code Drawer In JavaUsing Barcode encoder for Java Control to generate, create bar code image in Java applications.

Mathematica computes derivatives of combinations of functions, sums, differences, products, quotients, and composites by memorizing the various rules. If we do not define the functions, we can see what the rules are.

Generating DataMatrix In NoneUsing Barcode creator for Font Control to generate, create Data Matrix image in Font applications.

EAN / UCC - 13 Creator In NoneUsing Barcode printer for Font Control to generate, create UCC - 12 image in Font applications.

EXAMPLE 12

UPC Symbol Generation In .NETUsing Barcode generation for VS .NET Control to generate, create UPC-A Supplement 2 image in VS .NET applications.

Read USS Code 39 In Visual Basic .NETUsing Barcode scanner for Visual Studio .NET Control to read, scan read, scan image in .NET framework applications.

Clear[f, g] D[f[x] + g[x], x] f '[x] + g'[x] D[f[x] g[x], x] g[x] f'[x] + f[x] g'[x] D[f[x]/g[x], x]//Together g[x]f'[x] f[x]g'[x] 2 g[x] D[f[g[x]], x] f '[g[x]] g'[x]

Quotient rule. The derivative of a sum is the sum of the derivatives of its terms. This is the familiar product rule.

Chain rule.

We can use Mathematica to investigate some basic theory from a graphical perspective. Rolle s Theorem guarantees, under certain conditions, the existence of a point where the derivative of a function is 0: Let f be continuous on the closed interval [a, b] and differentiable on the open interval (a, b) and suppose f(a) = f(b) = 0. Then there exists a number, c, between a and b, such that f ' (c) = 0. In other words, if a smooth (differentiable) function vanishes (has a value of 0) at two distinct locations, its derivative must vanish somewhere in between.

EXAMPLE 13 Show that the function f ( x ) = ( x 3 + 2 x 2 + 15 x + 2)sin x satisfies Rolle s Theorem on the interval

[0, 1] and find the value of c referred to in the theorem. Since f is the product of a polynomial and a trigonometric sine function, f is continuous and differentiable everywhere. f[x_]=(x3 + 2 x2 + 15 x + 2) Sin[o x]; f[0] 0 f[1] 0 FindRoot[f'[c] 0, {c, 0.5}]

0.640241}

We used 0.5 as our initial guess since it is halfway between 0 and 1.

Plot[{f[x], f[.640241]}, {x, 0, 1}]

10 8 6 4 2

Differential Calculus

The Mean Value Theorem is similar to Rolle s Theorem and does not require f to be 0 at each endpoint of the interval: Let f be continuous on the closed interval [a, b] and differentiable on the open interval (a, b). Then there exists a number, c, between a and b such that f(b) f(a) = f ' (c) (b a). f (b) f (a) If we write the conclusion of the theorem in the form = f '(c), we see that the Mean Value b a Theorem guarantees the existence of a number, c, between a and b, such that the tangent line at (c, f(c)) is parallel to the line segment connecting the endpoints of the curve. Note: Rolle s Theorem and the Mean Value Theorem guarantee the existence of at least one number c. In actuality, there may be several.

EXAMPLE 14 Find the value(s), c, guaranteed by the Mean Value Theorem for the function f ( x ) =