Search any question & find its solution
Question:
Answered & Verified by Expert
If $x^{p} y^{q}=(x+y)^{p+q}$, then $\frac{d y}{d x}$ is equal to
Options:
Solution:
2069 Upvotes
Verified Answer
The correct answer is:
$y / x$
Given, $x^{p} y^{q}=(x+y)^{p+q}$
Taking log on both sides, we get
$$
\begin{array}{l}
\quad p \log x+q \log y=(p+q) \log (x+y) \\
\Rightarrow \quad \frac{p}{x}+\frac{q}{y} \frac{d y}{d x}=\frac{(p+q)}{(x+y)}\left(1+\frac{d y}{d x}\right) \\
\Rightarrow \quad\left(\frac{p}{x}-\frac{p+q}{x+y}\right)=\left(\frac{p+q}{x+y}-\frac{q}{y}\right) \frac{d y}{d x} \\
\Rightarrow \quad \frac{d y}{d x}=\frac{y}{x}
\end{array}
$$
Taking log on both sides, we get
$$
\begin{array}{l}
\quad p \log x+q \log y=(p+q) \log (x+y) \\
\Rightarrow \quad \frac{p}{x}+\frac{q}{y} \frac{d y}{d x}=\frac{(p+q)}{(x+y)}\left(1+\frac{d y}{d x}\right) \\
\Rightarrow \quad\left(\frac{p}{x}-\frac{p+q}{x+y}\right)=\left(\frac{p+q}{x+y}-\frac{q}{y}\right) \frac{d y}{d x} \\
\Rightarrow \quad \frac{d y}{d x}=\frac{y}{x}
\end{array}
$$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.