Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Question
**Problem:** Find the derivative of \(\ln\left(\frac{(4x + 12)^{10}}{x^8 + 9}\right)\). **Solution:** We need to find the derivative of the natural logarithm of the function: \[ f(x) = \ln\left(\frac{(4x + 12)^{10}}{x^8 + 9}\right) \] To make the differentiation easier, we can use the properties of logarithms to simplify the expression inside the logarithm: \[ \ln\left(\frac{(4x + 12)^{10}}{x^8 + 9}\right) = \ln((4x + 12)^{10}) - \ln(x^8 + 9) \] Using the property of logarithms \(\ln(a^b) = b \ln(a)\), we can further simplify: \[ \ln((4x + 12)^{10}) - \ln(x^8 + 9) = 10 \ln(4x + 12) - \ln(x^8 + 9) \] Now we can take the derivative of each term separately with respect to \(x\). 1. Derivative of \(10 \ln(4x + 12)\): \[ \frac{d}{dx} \left[ 10 \ln(4x + 12) \right] = 10 \cdot \frac{1}{4x + 12} \cdot \frac{d}{dx}(4x + 12) = 10 \cdot \frac{1}{4x + 12} \cdot 4 = \frac{40}{4x + 12} \] 2. Derivative of \(\ln(x^8 + 9)\): \[ \frac{d}{dx} \left[ \ln(x^8 + 9) \right] = \frac{1}{x^8 + 9} \cdot \frac{d}{dx}(x^8 + 9) = \frac{1}{x^8 + 9} \cdot 8x^7 = \frac{8x^7}{x^8 + 9} \] Combining both results, we get the derivative of \(f(x)\): \[ f'(x) = \frac{40}{
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Transcribed Image Text:**Problem:** Find the derivative of \(\ln\left(\frac{(4x + 12)^{10}}{x^8 + 9}\right)\). **Solution:** We need to find the derivative of the natural logarithm of the function: \[ f(x) = \ln\left(\frac{(4x + 12)^{10}}{x^8 + 9}\right) \] To make the differentiation easier, we can use the properties of logarithms to simplify the expression inside the logarithm: \[ \ln\left(\frac{(4x + 12)^{10}}{x^8 + 9}\right) = \ln((4x + 12)^{10}) - \ln(x^8 + 9) \] Using the property of logarithms \(\ln(a^b) = b \ln(a)\), we can further simplify: \[ \ln((4x + 12)^{10}) - \ln(x^8 + 9) = 10 \ln(4x + 12) - \ln(x^8 + 9) \] Now we can take the derivative of each term separately with respect to \(x\). 1. Derivative of \(10 \ln(4x + 12)\): \[ \frac{d}{dx} \left[ 10 \ln(4x + 12) \right] = 10 \cdot \frac{1}{4x + 12} \cdot \frac{d}{dx}(4x + 12) = 10 \cdot \frac{1}{4x + 12} \cdot 4 = \frac{40}{4x + 12} \] 2. Derivative of \(\ln(x^8 + 9)\): \[ \frac{d}{dx} \left[ \ln(x^8 + 9) \right] = \frac{1}{x^8 + 9} \cdot \frac{d}{dx}(x^8 + 9) = \frac{1}{x^8 + 9} \cdot 8x^7 = \frac{8x^7}{x^8 + 9} \] Combining both results, we get the derivative of \(f(x)\): \[ f'(x) = \frac{40}{
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