If xy=1, prove that dy dx +y^2=0 - Vidyadip

Question

If $\text{xy}=1,$ prove that $\frac{\text{dy}}{\text{dx}}+\text{y}^2=0$

✓

Answer

Here, xy = 1 .....(i)
Differentiating with respect to x,
$\frac{\text{d}}{\text{dx}}(\text{xy})=\frac{\text{d}}{\text{dx}}(1)$
$\Rightarrow\text{x}\frac{\text{dy}}{\text{dx}}+\text{y}\frac{\text{d}}{\text{dx}}(\text{x})=0$
[Using product rule]
$ \Rightarrow\text{x}\frac{\text{dy}}{\text{dx}}+\text{y}(1)=0$
$\Big[\text{Put x}=\frac{1}{\text{y}}\text{ from equation (i)}\Big]$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=-\frac{\text{y}}{\text{x}}$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=-\frac{\text{y}}{\frac{1}{\text{y}}}$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=-\text{y}^2$
$\Rightarrow\frac{\text{dy}}{\text{dx}}+\text{y}^2=0$

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 "5 Marks Questions" from Differentiation→Differentiation — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Coloured balls are distributed in $3$ bags as shown in the following table :

Colour of balls
Bag	Black	White	Red
$I$	$1$	$2$	$3$
$1$	$2$	$4$	$1$
$III$	$4$	$5$	$3$

A bag is selected at random and then two balls are randomly drawn from the selected bag. The colours of the balls are black and red. What is the probability that ball drawn is from bag I?

If $\text{A} = \begin{bmatrix} 1 & 2 & 5 \\ 1 & -1 & -1 \\ 2 & 3 & -1 \end{bmatrix}$ find $A^{–1}$ and hence solve the system of equations $x + 2y + 5z = 10, x – y – z = – 2$ and $2x + 3y – z = – 11.$

Find the intervals in which the following functions are increasing or decreasing.
$\text{f}(\text{x})=5\text{x}^{\frac{3}{2}}-3\text{x}^{\frac{5}{2}},\text{x}>0$

A manufacturer makes two products $A$ and $B$. Product $A$ sells at $Rs.200$ each and takes $1/2$ hour to make. Product $B$ sells at $Rs. 300$ each and takes $1$ hour to make. There is a permanent order for $14$ of product $A$ and $16$ of product $B$. A working week consists of $40$ hours of production and weekly turnover must not be less than $Rs.10000$. If the profit on each of product $A$ is $Rs. 20$ and on product $B$ is $Rs. 30,$ then how many of each should be produced so that the profit is maximum. Also, find the maximum profit.

An urn contains 5 red and 2 blcak balls. Two balls are randomly drawn, without replacement. Let X represent the number of black balls drawn. What are the possible values of X? Is X a random variable? If yes, then find the mean and variance of X.

A pair of dice is thrown 4 times. If getting a doublet is considered a success, find the probability distribution of number of successes.

Differentiate $\sin^{-1}\sqrt{1-\text{x}^2}$ with respect to $\cos^{-1}\text{x},$ if
$\text{x}\in(0, 1)$

If $\cos\text{y}=\text{x}\cos(\text{a}+\text{y}),$ where $\cos\text{a}\neq\pm1,$ prove that $\frac{\text{dy}}{\text{dx}}=\frac{\cos^2(\text{a}+\text{y})}{\sin\text{a}}$

Evaluate the following integrals:
$\int\frac{\text{e}^{\text{x}}}{\text{x}}\Big\{\text{x}(\log\text{x})^2+2\log\text{x}\Big\}\text{dx}$

Show that the family of curves for which $\frac{\text{dy}}{dx}=\frac{\text{x}^2+\text{y}^2}{2\text{xy}},\text{is given by}\ \text{x}^2-\text{y}^2=\text{c}x.$