Question

if xy=4 prove that x(dy/dx+y^2)=3y

Answer 1

$\textbf{Answer :}$

Given, xy = 4

or, x = 4/y

Now, differentiating both sides with respect to x, we get

d/dx (x) = 4 d/dx (1/y)

or, 1 = - 4/y² dy/dx

or, 4 dy/dx + y² = 0

or, 4 dy/dx = - y²

or, dy/dx = - y²/4

Now, x (dy/dx + y²)

= x (- y²/4 + y²)

= x (3y²/4)

= xy × (3y/4)

= 4 × (3y/4), since xy = 4

= 3y

Hence, proved.

#$\textbf{MarkAsBrainliest}$

Answer 2

Answer:

Given, xy = 4

or, x = 4/y

Now, differentiating both sides with respect to x, we get

d/dx (x) = 4 d/dx (1/y)

or, 1 = - 4/y² dy/dx

or, 4 dy/dx + y² = 0

or, 4 dy/dx = - y²

or, dy/dx = - y²/4

Now, x (dy/dx + y²)

= x (- y²/4 + y²)

= x (3y²/4)

= xy × (3y/4)

= 4 × (3y/4), since xy = 4

= 3y