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Chapter 3: Problem 81

Use the properties of logarithms to simplify the following functions before computing \(f^{\prime}(x)\). $$f(x)=\ln \frac{(2 x-1)(x+2)^{3}}{(1-4 x)^{2}}$$

Short Answer

Expert verified
Question: Find the derivative of the function \(f(x) = \ln \frac{(2x - 1)(x + 2)^3}{(1-4x)^2}\). Solution: The derivative of the function \(f(x)\) is: $$f^\prime(x) = \frac{2}{2x-1} + \frac{3}{x+2} + \frac{8}{1-4x}$$

Step by step solution

01

Simplify the function using properties of logarithms

Recall that the properties of logarithms are: 1. \(\ln(ab) = \ln(a) + \ln(b)\) 2. \(\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\) 3. \(\ln(a^n) = n\ln(a)\) Use these properties to simplify the given function: $$f(x) = \ln \frac{(2x - 1)(x + 2)^3}{(1-4x)^2}$$ Using property 2: $$f(x) = \ln(2x - 1) + \ln (x + 2)^3 - \ln (1 - 4x)^2$$ Now, using property 3: $$f(x) = \ln(2x - 1) + 3\ln (x + 2) - 2\ln (1 - 4x)$$ The simplified function is: $$f(x) = \ln(2x - 1) + 3\ln (x + 2) - 2\ln (1 - 4x)$$
02

Compute the derivative of the simplified function

Now that we have a simplified version of the function, we can find the derivative using the chain rule. Remember that \(\frac{d}{dx}\ln(u) = \frac{u'}{u}\). Let's differentiate each term: $$f^\prime(x) = \frac{d}{dx}\left(\ln(2x-1)\right) + \frac{d}{dx}\left(3\ln(x+2)\right) - \frac{d}{dx}\left(2\ln(1-4x)\right)$$ Applying the chain rule, we get: $$f^\prime(x) = \frac{2}{2x-1} + 3\frac{1}{x+2} - 2\frac{-4}{1-4x}$$
03

Simplify the derivative

Let's simplify the resulting expression to arrive at the final form of the derivative: $$f^\prime(x) = \frac{2}{2x-1} + \frac{3}{x+2} + \frac{8}{1-4x}$$ So the derivative of the simplified function \(f(x)\) is: $$f^\prime(x) = \frac{2}{2x-1} + \frac{3}{x+2} + \frac{8}{1-4x}$$

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