Derivative using Logarithmic Differentiation

Problem 110: Find the derivative of $f(x)$ using logarithmic differentiation when $f(x) = \frac{(x+1)^4}{(2x-5)^8}$ .

Solution: We will take the derivative of $f(x)$ using logarithmic properties. That is,

(1) $\begin{align*} f(x) & = \frac{(x+1)^4}{(2x-5)^8} \\ \ln(f(x)) & = \ln\left( \frac{(x+1)^4}{(2x-5)^8} \right) \\ \ln(f(x)) & = 4 \ln(x+1) - 8 \ln(2x-5). \quad \text{(property of logarithm)}. \end{align*}$

Differentiating both side we get

(2) $\begin{align*}\frac{1}{f(x)} \cdot f'(x) & = \frac{4}{x+1} - \frac{8}{2x-5} \cdot 2 \\ f'(x) & = f(x) \cdot \left( \frac{4}{x+1} - \frac{16}{2x-5}\right) \\ & = \frac{(x+1)^4}{(2x-5)^8} \cdot \left(\frac{4}{x+1} - \frac{16}{2x-5}\right) . \end{align*}$

That is,

(3) $\begin{equation*} \boxed{f'(x) = \frac{4(x+1)^3}{(2x-5)^8} - \frac{16(x+1)^4}{(2x-5)^9}}. \end{equation*}$

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Derivative using Logarithmic Differentiation

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