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If y = log[42x(x2+52x3-4)32], find dydx - Mathematics and Statistics

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Question

If y = $\log [4^{2 x} {(\frac{x^{2} + 5}{\sqrt{2 x^{3} - 4}})}^{\frac{3}{2}}]$ , find $\frac{d y}{d x}$

Sum

Solution

y = $\log [4^{2 x} {(\frac{x^{2} + 5}{\sqrt{2 x^{3} - 4}})}^{\frac{3}{2}}]$

= ${\log 4}^{2 x} + {\log (\frac{x^{2} + 5}{\sqrt{2 x^{3} - 4}})}^{\frac{3}{2}}$

= $2 x \log 4 + \frac{3}{2} [\frac{\log (x^{2} + 5)}{\sqrt{2 x^{3} - 4}}]$

= $2 x \log 4 + \frac{3}{2} [\log (x^{2} + 5) - \log \sqrt{2^{3} - 4}]$

= $2 x \log 4 + \frac{3}{2} [\log (x^{2} + 5) - \frac{3}{4} \log (2 x^{3} - 4)]$

Differentiating w. r. t. x, we get

$\frac{d y}{d x} = \frac{d}{d x} [2 x \log 4 + \frac{3}{2} \log (x^{2} + 5) - \frac{3}{4} \log (2 x^{3} - 4)]$

= $2 \log 4 \cdot \frac{d}{d x} (x) + \frac{3}{2} \cdot \frac{d}{d x} [\log (x^{2} + 5)] - \frac{3}{4} \cdot \frac{d}{d x} [\log (2 x^{3} - 4)]$

= $2 \log 4 \cdot 1 + \frac{3}{\cdot} \frac{1}{x^{2} + 5} \cdot \frac{d}{d x} (x^{2} + 5) - \frac{3}{4} \cdot \frac{1}{2 x^{3} - 4} \cdot \frac{d}{d x} (2 x^{3} - 4)$

= $2 \log 4 + \frac{3}{2} \cdot \frac{1}{x^{2} + 5} \cdot 2 x - \frac{3}{4} \cdot \frac{1}{2 x^{3} - 4} \cdot 6 x^{2}$

∴ $\frac{d y}{d x} = 2 \log 4 + \frac{3 x}{x^{2} + 5} - \frac{9 x^{2}}{2 (2 x^{3} - 4)}$

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Logarithmic Differentiation

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Chapter 2.1: Differentiation - Short Answers II

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SCERT Maharashtra Mathematics and Statistics (Arts and Science) [English] 12 Standard HSC

Chapter 2.1 Differentiation
Short Answers II | Q 2

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