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barcode in ssrs 2008 In addition, an expression such as 5 the new de nition. in Visual Studio .NET
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Recognizing QRCode In .NET Using Barcode scanner for .NET Control to read, scan read, scan image in .NET framework applications. Create Bar Code In Visual Studio .NET Using Barcode drawer for VS .NET Control to generate, create barcode image in .NET framework applications. De nition: ax is the unique positive real number such that ln ax = x ln a (35.1) Decoding Bar Code In VS .NET Using Barcode decoder for .NET Control to read, scan read, scan image in .NET applications. Painting QR Code In C# Using Barcode drawer for .NET framework Control to generate, create Quick Response Code image in Visual Studio .NET applications. To see that this de nition makes sense, observe that the equation ln y = x ln a must have exactly one positive solution y for each real number x. This follows from the fact that ln y is an increasing function with domain (0, + ) and range the set of all real numbers. (See the graph of the ln function in Fig. 342.) Make Quick Response Code In Visual Studio .NET Using Barcode printer for ASP.NET Control to generate, create QR Code JIS X 0510 image in ASP.NET applications. Quick Response Code Generator In VB.NET Using Barcode generation for .NET framework Control to generate, create QR Code JIS X 0510 image in VS .NET applications. PROPERTIES OF ax It is shown in Problem 35.4 that the function ax possesses all the standard properties of powers: GS1 DataBar Expanded Generator In Visual Studio .NET Using Barcode encoder for Visual Studio .NET Control to generate, create GS1 DataBar image in .NET applications. USS Code 128 Maker In .NET Framework Using Barcode creator for .NET Control to generate, create Code 128C image in .NET framework applications. (I) a0 = 1. (II) a1 = a. (III) au+v = au av . (IV) au v = (V) a v au . av 1 = v. a
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Bar Code Generation In .NET Framework Using Barcode maker for ASP.NET Control to generate, create barcode image in ASP.NET applications. Paint Barcode In C# Using Barcode creation for Visual Studio .NET Control to generate, create barcode image in VS .NET applications. CHAP. 35] EAN13 Supplement 5 Encoder In Java Using Barcode generation for Java Control to generate, create European Article Number 13 image in Java applications. EAN128 Generator In Visual C# Using Barcode creation for Visual Studio .NET Control to generate, create GTIN  128 image in .NET applications. EXPONENTIAL FUNCTIONS
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[CHAP. 35
In Problem 35.9 it is shown that ex is differentiable and that: Theorem 35.3: Dx ex = ex . Thus, ex has the property of being its own derivative. All constant multiples Cex share this property, since Dx Cex = C Dx ex = Cex . Problem 35.28 shows that these are the only functions with this property. From Theorem 35.3, we have ex dx = ex + C Knowing the derivative of ex , we obtain from Theorem 35.2 Dx ax = Dx (ex ln a ) = ex ln a Dx x ln a =e This proves: Theorem 35.4: Dx ax = ln a ax ax dx = ax +C ln a x ln a
(35.3) [by the chain rule] ln a = a ln a
or, in terms of antiderivatives, (35.4) We know that Dx x r = rx r 1 for any rational number r. Now the formula can be extended to arbitrary exponents. If, in Theorem 35.2, we replace x by r and a by x (thereby making x positive), we get x r = er ln x . Hence, Dx x r = Dx er ln x = er ln x Dx r ln x = er ln x r Thus, we have the following: Theorem 35.5: For any real number r and all positive x, Dx x r = rx r 1 1 x = xr r 1 x = rx r 1 [by the chain rule]

