Question

if y= cos inverse x [cosx+sin x] find dy/dx​

Answer 1

Heya!!

Before Differentiation let's make this function an easier one.

Cos x + Sin x =

√2 { Cos 45 × Cos x + Sin 45 × Sin x }

=>

√2 { Cos ( 45 - x ) }

=>

√2 { Cos ( x - 45 ) }

Becoz Cos (-Theta ) = Cos ( Theta )

=>

y = Cos-¹ { √2 Cos ( x - 45 ) }

Now, put √2 Cos ( x - 45 ) = z

dz/dx = -√2 Sin ( x - 45 ) ... equation ( i )

y = Cos-¹ ( z )

dy/dz = -1/ √ ( 1 - z² ) .... equation (ii )

Now, Multiple both the equations.

dy/dx =

√2 Sin ( x - 45 )
_________________
√ ( 1- 2 Cos² ( x - 45 )






dy/dx =

√2 Sin ( x - 45 )
___________________
√ ( 1 - 2 Cos² ( x - 45 )