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Chapter 3: Problem 5

State the derivative rule for the logarithmic function \(f(x)=\log _{b} x .\) How does it differ from the derivative formula for \(\ln x ?\)

Short Answer

Expert verified
Answer: The main difference between the derivative formulas for a logarithmic function with an arbitrary base b and the natural logarithm function is the presence of the factor ln b in the denominator of the formula for the logarithmic function with base b. This factor compensates for the change in base from e to b in the logarithmic function and reflects the change in slope for logarithmic functions with different bases, while the natural logarithm function does not require this factor.

Step by step solution

01

Rule for Derivative of the Logarithmic Function

The derivative rule for the logarithmic function with an arbitrary base \(b > 0\) and \(b \neq 1\), denoted as \(f(x) = \log_b x\), can be found using the properties of logarithms: $$\frac{d}{dx}(\log_b x) = \frac{1}{x\ln b}$$ The formula above represents the derivative of a function \(f(x) = \log_b x\), in which the base of the logarithm is \(b\).
02

Derivative Formula for \(\ln x\)

The derivative formula for the natural logarithm of \(x\), denoted as \(\ln x\), can be found by using the natural logarithm base \(e\). Here, \(b=e\): $$\frac{d}{dx}(\ln x) = \frac{1}{x}$$ This formula represents the derivative of a function \(f(x) = \ln x\), where the base of the logarithm is \(e\).
03

Comparing the Derivative Formulas

We can now compare the two formulas to see how they differ: 1. Logarithmic Function with Base \(b\): \(\frac{d}{dx}(\log_b x) = \frac{1}{x\ln b}\) 2. Natural Logarithm Function: \(\frac{d}{dx}(\ln x) = \frac{1}{x}\) The main difference between the two formulas lies in the factor of \(\ln b\) in the denominator of the first formula. The logarithmic function with base \(b\) (if \(b\neq e\)) has a changing factor depending on the base value, while the natural logarithm function, with base \(e\), does not require this factor. In other words, the presence of \(\ln b\) in the denominator of the first derivative formula compensates for the change in base from \(e\) to \(b\) in the logarithmic function and reflects the change in slope for logarithmic functions with different bases.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logarithmic Differentiation
Logarithmic differentiation is a powerful mathematical tool used to differentiate functions that are challenging to handle with standard rules of differentiation. It becomes particularly useful when dealing with products, quotients, or powers of functions where the argument of the logarithmic function is more complex than just a simple variable like 'x'.

For example, if we have a function of the form h(x) = f(x)g(x), taking the natural logarithm of both sides gives us ln(h(x)) = ln(f(x)g(x)). Applying properties of logarithms, we can simplify this to ln(h(x)) = ln(f(x)) + ln(g(x)). When we differentiate both sides using the chain rule and properties of logarithms, we find a simpler way to find the derivative of the original function.
Natural Logarithm Derivative
The derivative of the natural logarithm function is a cornerstone in calculus because of its elegant simplicity and application in various disciplines. With the natural logarithm function \(f(x) = \ln(x)\), the derivative is simply \(\frac{d}{dx}(\ln x) = \frac{1}{x}\).

This formula tells us that the slope of the tangent line to the curve \(y = \ln(x)\) at any point is the reciprocal of the x-coordinate at that point. The natural logarithm derivative comes into play in numerous scenarios, such as when finding the time constant in physics or the rate of growth or decay in biology and economics.
Properties of Logarithms
Properties of logarithms are the rules that allow us to manipulate logarithmic expressions to simplify complex problems. These properties stem from the definition of logarithms as the inverse functions of exponentials. Some of the essential properties include:
  • The product rule: \(\log_b(mn) = \log_b(m) + \log_b(n)\)
  • The quotient rule: \(\log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n)\)
  • The power rule: \(\log_b(m^n) = n\log_b(m)\)
  • Change of base formula: \(\log_b(m) = \frac{\log_k(m)}{\log_k(b)}\), for any positive base \($k\)

These properties are not just theoretical concepts; they are practical tools for solving calculus problems involving logarithmic functions.
Calculus
Calculus is a branch of mathematics that studies change and motion through derivatives and integrals. Derivatives describe how a quantity changes over time, quite literally the 'rate of change', whereas integrals provide the accumulated value, like the area under a curve.

Calculus is fundamental in science and engineering and underpins many phenomena in physics, biology, economics, and beyond. When applied to logarithmic functions, calculus helps us understand growth rates and decay, optimize functions, and solve difficult equations, further highlighting the interconnectedness of mathematical concepts.

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Most popular questions from this chapter

Calculate the derivative of the following functions (i) using the fact that \(b^{x}=e^{x \ln b}\) and (ii) by using logarithmic differentiation. Verify that both answers are the same. $$y=\left(x^{2}+1\right)^{x}$$

Jean and Juan run a one-lap race on a circular track. Their angular positions on the track during the race are given by the functions \(\theta(t)\) and \(\varphi(t),\) respectively, where \(0 \leq t \leq 4\) and \(t\) is measured in minutes (see figure). These angles are measured in radians, where \(\theta=\varphi=0\) represent the starting position and \(\theta=\varphi=2 \pi\) represent the finish position. The angular velocities of the runners are \(\theta^{\prime}(t)\) and \(\varphi^{\prime}(t)\). a. Compare in words the angular velocity of the two runners and the progress of the race. b. Which runner has the greater average angular velocity? c. Who wins the race? d. Jean's position is given by \(\theta(t)=\pi t^{2} / 8 .\) What is her angular velocity at \(t=2\) and at what time is her angular velocity the greatest? e. Juan's position is given by \(\varphi(t)=\pi t(8-t) / 8 .\) What is his angular velocity at \(t=2\) and at what time is his angular velocity the greatest?

a. Differentiate both sides of the identity \(\cos 2 t=\cos ^{2} t-\sin ^{2} t\) to prove that \(\sin 2 t=2 \sin t \cos t\). b. Verify that you obtain the same identity for sin \(2 t\) as in part (a) if you differentiate the identity \(\cos 2 t=2 \cos ^{2} t-1\). c. Differentiate both sides of the identity \(\sin 2 t=2 \sin t \cos t\) to prove that \(\cos 2 t=\cos ^{2} t-\sin ^{2} t\).

Quotient Rule for the second derivative Assuming the first and second derivatives of \(f\) and \(g\) exist at \(x,\) find a formula for \(\frac{d^{2}}{d x^{2}}\left(\frac{f(x)}{g(x)}\right)\)

Calculating limits exactly Use the definition of the derivative to evaluate the following limits. $$\lim _{x \rightarrow e} \frac{\ln x-1}{x-e}$$
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