Question

if sin y=xsin( a+y) then dy/ dx = ?

Answer 1

The Proof is given in the Explanation.

Explanation:

siny=xsin(a+y).

∴x=sinysin(a+y).

Differentiating w.r.t. y, using the Quotient Rule,we have,

dxdy=sin(a+y)⋅ddy{siny}−siny⋅ddx{sin(a+y)}sin2(a+y),

=sin(a+y)cosy−sinycos(a+y)⋅ddy(a+y)sin2(a+y),...[The Chain Rule],

=sin(a+y)cosy−sinycos(a+y)sin2(a+y),

=sin{(a+y)−y}sin2(a+y),

=sinasin2(a+y).

⇒dydx=1dxdy=sin2(a+y)sina

Answer 2

here siny=xsin (a+y)
now X= sinY/sin (a+y)
isko differentiate kro LHS one aajayega
RHS ko product rule se solve krne pe kuch aisa ayega
1 = cosydy/dx . sin (a+y) - siny. cos(a+y)dy/dx ÷ sin^2(a+y)
now we get sin^2 (a+y)= cosydy/dx.sin (a+y) - cos(a+y)dy/dx.siny
ab RHS mein dy/dx common lelo
to ayega
cosy.sin (a+y) - siny.cos (a+y)
ye sidhe sin(x-y) wale formula pe hai to ye simplify krke
sinAajayega ab
LHS hai sin^2 (a+y)
or RHS sinA dy/dx
LHS=RHS kro
sin^2 (a+y)= sina dy/dx ayega
phir dy/dx = sin^2(a+y)/sina
answer