Question

Determine whether the following statements are true and give an explanation or counterexample. a. The derivative of \(\log _{2} 9\) is \(1 /(9 \ln 2)\) b. \(\ln (x+1)+\ln (x-1)=\ln \left(x^{2}-1\right),\) for all \(x\) c. The exponential function \(2^{x+1}\) can be written in base \(e\) as \(e^{2 \ln (x+1)}\) d. \(\frac{d}{d x}(\sqrt{2} x)=x \sqrt{2}^{x-1} . \quad\) e. \(\frac{d}{d x}\left(x^{\sqrt{2}}\right)=\sqrt{2} x^{\sqrt{2}-1}\) f. \(\quad(4 x+1)^{\ln x}=x^{\ln (4 x+1)}\)

Short answer

Question: Determine if the following statements are true or false, and provide an explanation or counterexample. a) The derivative of \(\log_2 9\) is \(1/(9 \ln 2)\). b) \(\ln (x+1)+\ln (x-1)=\ln \left(x^{2}-1\right)\). c) \(2^{x+1}\) can be written as \(e^{2 \ln (x+1)}\). d) The derivative of \(\sqrt{2}x\) is \(x\sqrt{2}^{x-1}\). e) The derivative of \(x^{\sqrt{2}}\) is \(\sqrt{2}x^{\sqrt{2}-1}\). f) \((4x+1)^{\ln x}=x^{\ln(4x+1)}\). Answer: a) False. The derivative of \(\log_2 9\) is 0 since it is a constant. b) True for all \(x\) except \(x = -1\) and \(x = 1\), where the logarithm is undefined. c) False. \(2^{x+1}\) can be written as \(e^{(x+1) \ln 2}\), which is different from \(e^{2\ln(x+1)}\). d) False. The derivative of \(\sqrt{2}x\) is \(\sqrt{2}\), not \(x\sqrt{2}^{x-1}\). e) True. The derivative of \(x^{\sqrt{2}}\) is \(\sqrt{2}x^{\sqrt{2}-1}\). f) True. \((4x+1)^{\ln x}=x^{\ln(4x+1)}\).

Step-by-step solution

Determine if the derivative of \(\log_2 9\) is \(1/(9 \ln 2)\)

The \(\log_2 9\) is a constant, so its derivative is equal to 0. So the statement is false. The correct answer would be 0. b.

Check if \(\ln (x+1)+\ln (x-1)=\ln \left(x^{2}-1\right)\)

We can use the logarithmic property \(\ln A + \ln B = \ln (AB)\): $$\ln(x+1) + \ln(x-1) = \ln((x+1)(x-1)).$$ We can simplify the right side like this: $$(x+1)(x-1) = x^2 - 1,$$ so the equation becomes $$\ln (x+1) + \ln (x-1) = \ln (x^2 - 1).$$ This statement is true for all \(x\) except for \(x = -1\) and \(x = 1\), where the logarithm is undefined. c.

Check if \(2^{x+1}\) can be written as \(e^{2 \ln (x+1)}\)

First, rewrite \(2^{x+1}\) using the exponential property: $$2^{x+1} = e^{\ln(2^{x+1})}.$$ Now, apply the logarithmic property \(\ln A^B = B \ln A\): $$e^{\ln(2^{x+1})} = e^{(x+1) \ln 2},$$ which is different from the given expression \(e^{2\ln(x+1)}\). So, this statement is false. d.

Compute the derivative of \(\sqrt{2}x\)

Calculate the derivative with respect to x: $$\frac{d}{dx} (\sqrt{2}x) = \sqrt{2},$$ which is different from the given expression \(x\sqrt{2}^{x-1}\). This statement is false. e.

Compute the derivative of \(x^{\sqrt{2}}\)

Calculate the derivative with respect to x: $$\frac{d}{dx} \left(x^{\sqrt{2}}\right) = \sqrt{2}x^{\sqrt{2}-1},$$ The given expression is the same, so this statement is true. f.

Check if \((4x+1)^{\ln x}=x^{\ln(4x+1)}\)

Take the natural logarithm of both sides: $$\ln((4x+1)^{\ln x}) = \ln(x^{\ln(4x+1)}).$$ Now apply the logarithmic property \(\ln A^B = B \ln A\): $$\ln x \ln(4x+1) = \ln(4x+1) \ln x.$$ This equation is true, so the original statement is also true.

Key concepts

Derivative

The derivative is a central concept in calculus. It represents the rate at which a function is changing at any given point. When we say the derivative of a function, we are referring to its instantaneous rate of change. A classic example is the derivative of a constant function, which is zero. This is because constant values do not change, hence their rate of change is zero. In the original exercise, the statement was incorrect because it suggested that the logarithm to a constant value has a derivative other than zero. Remember, for any constant function of the form \(f(x) = c\), where \(c\) is a constant, the derivative \(f'(x) = 0\). This concept is crucial for understanding how derivatives reflect the behavior of functions.

Exponential Function

Exponential functions involve expressions with constants raised to the power of a variable. These functions have unique properties, particularly related to their bases and their growth rates. In calculus, working with exponential functions often involves rewriting them in different forms using natural exponential functions with base \(e\). The expression \(2^{x+1}\) can be rewritten as \(e^{(x+1)\ln 2}\), showcasing its dependency on the natural logarithm. The base \(e\) is preferred in many mathematical contexts because it simplifies differentiation and integration processes as it directly relates to the natural logarithm. A key misunderstanding in conversions like these can lead to incorrect forms, like the one seen in the provided exercise.

Logarithmic Properties

Logarithmic properties are essential when manipulating expressions involving logarithms. One fundamental property is that the logarithm of a product is the sum of the logarithms: \(\ln A + \ln B = \ln (AB)\). This property helps in simplifying expressions, as seen in the exercise where \(\ln(x+1) + \ln(x-1)\) was correctly simplified to \(\ln(x^2 - 1)\). Such properties apply universally to valid inputs of logarithmic functions, but keep in mind the domain restrictions: logarithms are undefined for non-positive values. Understanding these rules allows for efficient manipulation and simplification of log expressions in calculus.

Natural Logarithm

The natural logarithm, denoted as \(\ln\), is a logarithm with base \(e\), an irrational and transcendental number approximately equal to 2.718. The natural logarithm is heavily used in calculus due to its convenience in differentiation. The derivative of \(\ln x\) is \(1/x\), which simplifies many calculations. In the exercise, using properties of natural logarithms helped in verifying the truths of certain statements, as natural logs simplify the expression of exponential and power functions. When dealing with natural logarithms, understanding their properties, such as \(\ln(e^x) = x\) and \(e^{\ln x} = x\), enables solving and simplifying complex calculus problems effectively.