Question

DIFFERENTIAL CALCULUS
If y = x sin^-1 x / (√1-x^2) ,prove that (1-x^2)dy/dx =x + y/x
(with steps)

Answer 1

,hope this helps you........

Answer 2

(1 - x²)dy/dx = x + y/x   if  y = x sin⁻¹ x / (√1-x²)

Step-by-step explanation:

y = x sin⁻¹ x / (√1-x²)

Let say x = Sinα

=> y = Sinα * α /(√1-Sin²α)

=> y = Sinα * α /(√Cos²α)

=> y = Sinα * α /Cosα

=> y = α tanα

=> dy/dα  =  tanα  + α(sec²α)

=> dy/dα  =  tanα  + y(sec²α)/tanα

=> dy/dα  =  (tan²α  + y(tan²α  + 1))/tanα

x = Sinα

=> dx/dα = Cosα

=> (dy/dα)/(dx/dα)  =  (tan²α  + y(tan²α  + 1))/(tanα(Cosα))

=> dy/dx = (tan²α  + y(tan²α  + 1))/(Sinα)

x = Sinα

=> Cosα = √1 - x²

tanα  = x/√1 - x²

tan²α  = x²/(1 - x²)

dy/dx =  (x²/(1 - x² )  +  y( x²/(1 - x²) + 1) )/x

dy/dx =  (x²/(1 - x² )  +  y( (x² + 1 - x²)/(1 - x²) )/x

dy/dx =  (x²/(1 - x² )  +  y/(1 - x²) )/x

dy/dx =  (x² +  y/(1 - x²) )/x

dy/dx =  (x² +  y )/(1 - x²)x

(1 - x²)dy/dx =  (x²  +  y )/x

(1 - x²)dy/dx = x + y/x

QED

Proved

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