CONTINUITY AND DIFFERENTIABILITY — Maths STD 12 Science — Question
Gujarat BoardEnglish MediumSTD 12 ScienceMathsCONTINUITY AND DIFFERENTIABILITY4 Marks
Question
If $\text{xy}=4,$ prove that $\text{x}\Big(\frac{\text{dy}}{\text{dx}}+\text{y}^2\Big)=3\text{y}$
✓
Answer
We have, $\text{xy}=4$ $\Rightarrow\text{y}=\frac{4}{\text{x}}$ Differentiating with respect to x, $\frac{\text{dy}}{\text{dx}}=\frac{\text{d}}{\text{dx}}\big(\frac{4}{\text{x}}\big)$ $\Rightarrow\frac{\text{dy}}{\text{dx}}=4\frac{\text{d}}{\text{dx}}\big(\text{x}^{-1}\big)$ $\Rightarrow\frac{\text{dy}}{\text{dx}}=4(-1\times\text{x}^{-1-1})$ $\Rightarrow\frac{\text{dy}}{\text{dx}}=4\Big(-\frac{1}{\text{x}^2}\Big)$ $\Rightarrow\frac{\text{dy}}{\text{dx}}=\frac{-4}{\text{x}^2}$ $\Rightarrow\frac{\text{dy}}{\text{dx}}=-\frac{4}{\big(\frac{4}{\text{y}}\big)^2}\ \Big[\because\text{x}=\frac{4}{\text{y}}\Big]$ $\Rightarrow\frac{\text{dy}}{\text{dx}}=-\frac{4\text{y}^2}{16}$ $\Rightarrow\frac{\text{dy}}{\text{dx}}=-\frac{\text{y}^2}{4}$ $\Rightarrow4\frac{\text{dy}}{\text{dx}}=-\text{y}^2$ $\Rightarrow4\frac{\text{dy}}{\text{dx}}+4\text{y}^2=3\text{y}^2$ $\Rightarrow4\Big(\frac{\text{dy}}{\text{dx}}+\text{y}^2\Big)=3\text{y}^2$ Dividing both side by x, $\Rightarrow\frac{4}{\text{x}}\Big(\frac{\text{dy}}{\text{dx}}+\text{y}^2\Big)=\frac{3\text{y}^2}{\text{x}}$ $\Rightarrow\text{y}\Big(\frac{\text{dy}}{\text{dx}}+\text{y}^2\Big)=\frac{3\text{y}^2}{\text{y}}$ $\Rightarrow\text{x}\Big(\frac{\text{dy}}{\text{dx}}+\text{y}^2\Big)=\frac{3\text{y}^2}{\text{y}}$ $\Rightarrow\text{x}\Big(\frac{\text{dy}}{\text{dx}}+\text{y}^2\Big)=3\text{y}$
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