Limits, Continuity and Differentiability
80277
If \(y=\left(\frac{x^{2}}{x+1}\right)^{x}\) and \(\frac{d y}{d x}=y\left[g(x)+\log \left(\frac{x^{2}}{x+1}\right)\right]\), then \(g(x)=\)
1 \(\frac{x^{2}}{x+1}\)
2 \(\frac{x-1}{x+2}\)
3 \(\frac{x+2}{x+1}\)
4 \(x \log \left(\frac{x^{2}}{x+1}\right)\)
Explanation:
(C) : Given,
\(y=\left(\frac{x^{2}}{x+1}\right)^{x}\)
Taking log both sides,
\(\log \mathrm{y}=\log \left(\frac{\mathrm{x}^{2}}{\mathrm{x}+1}\right)^{\mathrm{x}} \Rightarrow \log \mathrm{y}=\mathrm{x} \log \left(\frac{\mathrm{x}^{2}}{\mathrm{x}+1}\right)\)
Differentiating both sides with respect to \(\mathrm{x}\)
\(\frac{1}{y} \frac{d y}{d x}= {\left[x \frac{d}{d x} \log \left(\frac{x^{2}}{x+1}\right)+\log \left(\frac{x^{2}}{x+1}\right) \cdot \frac{d}{d x}(x)\right] }\)
\(\frac{1}{y} \frac{d y}{d x}=x \times \frac{1}{\frac{x^{2}}{x}} \cdot \frac{d}{d x}\left(\frac{x^{2}}{x+1}\right)+\log \left(\frac{x^{2}}{x+1}\right) \times 1\)
\(\frac{1}{y} \frac{d y}{d x}=\frac{x(x+1)}{x^{2}}\left[\frac{(x+1)(2 x)-x^{2}(1+0)}{(x+1)^{2}}\right]+\log \left(\frac{x^{2}}{x+1}\right)\)
\(\frac{d y}{d x}=y\left[\frac{x+1}{x}\left(\frac{2 x^{2}+2 x-x^{2}}{(x+1)^{2}}\right)+\log \left(\frac{x^{2}}{x+1}\right)\right]\)
\(\frac{d y}{d x}=y\left[\left(\frac{x^{2}+2 x}{x(x+1)}\right)+\log \left(\frac{x^{2}}{x+1}\right)\right]\)
\(\frac{d y}{d x}=y\left[\frac{x(x+2)}{x(x+1)}+\log \left(\frac{x^{2}}{x+1}\right)\right]\)
\(\frac{d y}{d x}=y\left[\frac{x+2}{x+1}+\log \left(\frac{x^{2}}{x+1}\right)\right]\)
Therefore,
\(\frac{d y}{d x}=y\left[g(x)+\log \left(\frac{x^{2}}{x+1}\right)\right]\)
Putting the value of \(\frac{d y}{d x}\) in above equation
Then,
\(y\left[\frac{x+2}{x+1}+\log \left(\frac{x^{2}}{x+1}\right)\right]=y\left[g(x)+\log \left(\frac{x^{2}}{x+1}\right)\right]\)
\(\frac{x+2}{x+1}+\log \left(\frac{x^{2}}{x+1}\right)=g(x)+\log \left(\frac{x^{2}}{x+1}\right)\)
\(\mathrm{g}(\mathrm{x})=\frac{\mathrm{x}+2}{\mathrm{x}+1}\)
