If xy=4, prove that x ( dy dx +y^2 )=3y - Vidyadip

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,
$\Rightarrow\frac{4}{\text{x}}\Big(\frac{\text{dy}}{\text{dx}}+\text{y}^2\Big)=\frac{3\text{y}^2}{\text{x}}$
$\Rightarrow\text{y}\Big(\frac{\text{dy}}{\text{dx}}+\text{y}^2\Big)=\frac{3\text{y}^2}{\text{y}}$
$\Rightarrow\text{x}\Big(\frac{\text{dy}}{\text{dx}}+\text{y}^2\Big)=\frac{3\text{y}^2}{\text{y}}$
$\Rightarrow\text{x}\Big(\frac{\text{dy}}{\text{dx}}+\text{y}^2\Big)=3\text{y}$

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 "4 Marks" from Differentiation→Differentiation — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Find the equation of the plane passing through the point (-1, 3, 2) and perpendicular to each of the planes $\text{x + 2y +3z = 5 and 3x + 3y + z} = 0$

find the area of the region included between the parabola y² = x and the line x + y = 2.

Show that for any two vectors $\vec a $ and $\vec b$ , we always have $|\vec{a}+\vec{b}| \leq|\vec{a}|+|\vec{b}|$ (triangle inequality).

Differentiate the following functions with respect to x:
$\text{e}^{\text{ax}}\sec\text{x}\tan2\text{x}$

A manufacturer has three machines installed in his factory. machines I and II are capable of being operated for at most 12 hours whereas Machine III must operate at least for 5 hours a day. He produces only two items, each requiring the use of three machines. The number of hours required for producing one unit each of the items on the three machines is given in the following table:

Item	Number of hours required by the machine
	I	II	III
A	1	2	1
B	2	1	$\frac{5}{4}$

He makes a profit of Rs. 6.00 on item A and Rs. 4.00 on item B. Assuming that he can sell all that he produces, how many of each item should he produces so as to maximize his profit? Determine his maximum profit. Formulate this LPP mathematically and then solve it.

The direction ratios of the perpendicular from the origin to a plane are 12, -3, 4 and the length of the perpendicular is 5. Find the equation of the plane.

Using integration find the area of the region bounded by the curve $\text{y}=\sqrt{4-\text{x}^2},\text{ x}^2+\text{y}^2-4\text{x}=0$ and the x-axis.

Find the area bounded by the parabola y² = 4x and the line y = 2x - 4:
By using horizontal strips.

Find the vector equation of the plane through the line of intersection of the planes x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane x - y + z = 0.

Verify Rolle's theorem for the following function on the indicated intervals
f(x) = x² -4x + 3 on [1, 3]