If x^p y^q=(x+y)^ p+q then prove that d y d x =y x - Vidyadip

Question

If $x^p y^q=(x+y)^{p+q}$ then prove that $\frac{d y}{d x}=\frac{y}{x}$

✓

Answer

$x^py^{q }= (x + y)^{p+q}$
Taking $\log$ both side
$p \log x + q \log y = (p + q) \log (x + y)$
Differentiating $w.r.t. x$
$ \frac{p}{x}+\frac{q}{y} \frac{d y}{d x}=\frac{p+q}{x+y}+\left(\frac{p+q}{x+y}\right) \frac{d y}{d x}$
$ \frac{q}{y} \frac{d y}{d x}-\left(\frac{p+q}{x+y}\right) \frac{d y}{d x}=\frac{p+q}{x+y}-\frac{p}{x}$
$ \left(\frac{q}{y}-\frac{p+q}{x+y}\right) \frac{d y}{d x}=\left(\frac{p+q}{x+y}-\frac{p}{x}\right)$
$ \left(\frac{q x-p y}{y}\right) \frac{d y}{d x}=\left(\frac{q x-p y}{x}\right)$
$ \frac{1}{y} \frac{d y}{d x}=\frac{1}{x}$
$ \frac{d y}{d x}=\frac{y}{x}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "Solve the Following Question.(3 Marks)" from Differentiation→Differentiation — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Find the volume of a tetrahedron whose vertices are A( -1, 2, 3) B(3, -2, 1), C (2, 1, 3) and D(-1, -2, 4).

For the following matrices verify the distributivity of matrix, multiplication over matrix addtion i.e., A(B + C) = AB + AC.
$\text{A}=\begin{bmatrix}1&-1\\0&2\end{bmatrix},\text{B}=\begin{bmatrix}-1&0\\2&1\end{bmatrix}$ and $\text{C}=\begin{bmatrix}0&1\\1&-1\end{bmatrix}$

Evaluate the following integrals:$\int\limits^{\frac{\pi}{2}}_{-\frac{\pi}{2}}\frac{-\frac{\pi}{2}}{\sqrt{\cos\text{x}\sin^2\text{x}}}\text{ dx}$

An open box is to be cut out of piece of square card coard of side $18 \mathrm{~cm}$ by cutting of equal squares from the corners and turning up the sides. Find the maximum volume of the box.

If $|x| \leq 1$, show that
$\sin \left(\cos ^{-1} x\right)=\cos \left(\sin ^{-1} x\right)$

Using vectors, find the value of $\lambda$ such that the points ($\lambda$, -10, 3), (1, -1, 3) and (3, 5, 3) are collinear.

If $\bar{c}=3 \bar{a}-2 \bar{b}$ prove that $[\bar{a} \bar{b} \bar{c}]=0$

Evaluate the following definite integrals:$\int_{4}^\limits{9}\frac{1}{\sqrt{\text{x}}}\text{ dx}$

Find $\frac{d y}{d x}$ if : $x=a(\theta+\sin \theta), y=a(1-\cos \theta)$

If $x < 0, y < 0$ such that $xy = 1$, then write the value of $\tan^{-1}x + \tan^{-1}y$.