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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 difference between these two derivative rules lies in the fact that the derivative of \(f(x) = \log_b x\) has an additional factor \(\ln b\) in the denominator, while the derivative of \(g(x) = \ln x\) does not. Specifically, \(f'(x) = \frac{1}{x \ln b}\) and \(g'(x) = \frac{1}{x}\).

Step by step solution

01

Derivative Rule for Logarithmic Function \(f(x) = \log_b x\)

To find the derivative of \(f(x) = \log_b x\), we'll use the chain rule and properties of logarithms. The chain rule states that if \(f(x) = u(v(x))\), then \(f'(x) = u'(v(x)) \cdot v'(x)\). In this case, \(u(v) = \log_b v\), so \(u'(v) = \frac{1}{v \ln b}\). The chain rule then gives us: $$f'(x) = \frac{1}{x \ln b}$$
02

Derivative Formula for Natural Logarithm \(g(x) = \ln x\)

The derivative formula for the natural logarithm function \(g(x) = \ln x\) is: $$g'(x) = \frac{1}{x}$$
03

Comparing the Derivative Rules

The derivative rule for the logarithmic function \(f(x) = \log_b x\) is: $$f'(x) = \frac{1}{x \ln b}$$ The derivative formula for the natural logarithm function \(g(x) = \ln x\) is: $$g'(x) = \frac{1}{x}$$ The difference between these two derivative rules lies in the fact that the derivative of \(f(x) = \log_b x\) has an additional factor \(\ln b\) in the denominator, while the derivative of \(g(x) = \ln x\) does not.

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