Q17 of 52 Page 1

If , show that .

Or


If xy-yx=ab, find .


Given that


Differentiating with respect to itself, we get





Simplifying, we get


x dx + y dy = x dy – y dx

or


dx(x + y)= dy(x-y)


Hence,


OR


Given that xy-yx=ab


Let u= xy and v= yx, then we get


u-v= ab


Differentiating with respect to x, we get


(Since ab is constant) …. (1)


u= xy


Taking log both sides



Differentiating with respect to x, we get



⇒


Hence, ….. (2)


v= yx


Taking log both sides



Differentiating with respect to x, we get





⇒


Hence, ……(3)


Substituting (2) and (3) in (1), we get



Simplifying, we get


More from this chapter

All 52 →
15

Show that the relation R on ℝ defined as R={(a , b):a ≤ b}, is reflexive, and transitive but not symmetric.

Or

Prove that the function f:ℕ→ℕ, defined by f(x)=x2+x+1 is one-one but not onto. Find inverse of f:ℕ→S, where S is range of f.

 16

Find the equation of the tangent to the curve which is parallel to the line 4x-2y+5=0. Also, write the equation of normal to the curve at the point of contact.

 18

If y=(sin-1x)2, prove that .

 19

Prove that , hence evaluate .