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Differentiation — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsDifferentiation3 Marks
Question
Diffrentiate the following w.r.t.x

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

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Answer

$\begin{aligned} & \quad \text { Let } y=\log \left[4^{2 x}\left(\frac{x^2+5}{\sqrt{2 x^3-4}}\right)^{\frac{3}{2}}\right] \\ & =\log 4^{2 x}+\log \left(\frac{x^2+5}{\sqrt{2 x^3-4}}\right)^{\frac{3}{2}} \\ & =2 x \log 4+\frac{3}{2} \log \left(\frac{x^2+5}{\sqrt{2 x^3-4}}\right) \\ & =2 x \log 4+\frac{3}{2}\left[\log \left(x^2+5\right)-\log \left(2 x^3-4\right)^{\frac{1}{2}}\right] \\ & =2 x \log 4+\frac{3}{2}\left[\log \left(x^2+5\right)-\frac{1}{2} \log \left(2 x^3-4\right)\right] \\ & =2 x \log 4+\frac{3}{2} \log \left(x^2+5\right)-\frac{3}{4} \log \left(2 x^3-4\right)\end{aligned}$

Differentiating w.r.t. $x$, we get

$\begin{aligned} & \frac{d y}{d x}=\frac{d}{d x}\left[2 x \log 4+\frac{3}{2} \log \left(x^2+5\right)-\frac{3}{4} \log \left(2 x^3-4\right)\right] \\ & =(2 \log 4) \frac{d}{d x}(x)+\frac{3}{2} \frac{d}{d x}\left[\log \left(x^2+5\right)\right]-\frac{3}{4} \frac{d}{d x}\left[\log \left(2 x^3-4\right)\right] \\ & =(2 \log 4) \times 1+\frac{3}{2} \times \frac{1}{x^2+5} \cdot \frac{d}{d x}\left(x^2+5\right)- \\ & =2 \log 4+\frac{3}{2\left(x^2+5\right)} \times(2 x+0)-\frac{3}{4\left(2 x^3-4\right)} \times \frac{1}{2 x^3-4} \cdot \frac{d}{d x}\left(2 x^3-4\right) \\ & =2 \log 4+\frac{3 x}{x^2+5}-\frac{\left.9 x^2-0\right)}{2\left(2 x^3-4\right)} .\end{aligned}$

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