Question

y=cos⁻¹ 1-x²/1+x²,Find the derivative.

Answer 1

it is given that y = cos^-1{(1 - x²)/(1 + x²)}

differentiate with respect to x,

dy/dx = d[cos^-1{(1 - x²)/(1 + x²)}]/dx

=- 1/√[1 - {(1 - x²)/(1 + x²)}²] × d{(1 - x²)/(1 + x²)}/dx

= -(1 + x²)/√{(1 + x²)² - (1 - x²)²} × [{(1 + x²) × (-2x) - (1 - x²) × 2x}/(1 + x²)²]

=- (1 + x²)/√{1 + x⁴ + 2x² - 1 - x⁴ + 2x²} × [(-2x - 2x³ -2x + 2x³)/(1 + x²)²]

=- (1 + x²) × (-4x)/{√(4x²) × (1 + x²)²}

= 4x/{2x(1 + x²)}

= 2/(1 + x²)

hence, derivative of y is y' = 2/(1 + x²)

Answer 2

HELLO DEAR,



GIVEN:-
Y = cos-¹{1 - x²}/{1 + x²}

put x = tant
=> t = tan-¹x


Y = cos-¹{(1 - tan²t)/(1 + tan²t)}

Y = cos-¹(cos2t)

Y = 2t

dy/dx = dy/dt × dt/dx

=> dy/dx = d(2t)/dt × d(tan-¹x)/dx

=> dy/dx = 2 × 1/(1 + x²)

=> dy/dx = 2/(1 + x²).



I HOPE ITS HELP YOU DEAR,
THANKS