AP EAMCET201822 Apr 2018Morning ShiftMathematicsDifferentiationActual
The derivative of f(x)=x^ ⁻¹ x with respect to g(x)= ⁻¹ ( 1 2 x^2-1 ) is
Options
- A1 2 1-x^2 x^ ⁻¹ x [ x 1+x^2 + ⁻¹ x x ]
- B- 1 2 1-x^2 x^ x^ -a⁻¹ [ ( ⁻¹ x )+x (1+x^2 ) ⁻¹ x ]
- C.-2 ⁻¹ x 1+x^2 + ⁻¹ x x ] 1-x^2
- D- 1 2 1-x^2 x^ x⁻¹ [ x 1+x^2 + ⁻¹ x x ]
Correct answer
D. - 1 2 1-x^2 x^ x⁻¹ [ x 1+x^2 + ⁻¹ x x ]
Step-by-step solution
We have, aligned f(x) & =x^ ⁻¹ x f(x) & = ⁻¹ x x aligned aligned & 1 f(x) d d x f(x)= 1 1+x^2 x+ ⁻¹ x x & d f(x) d x =x^ ⁻¹ x [ x 1+x^2 + ⁻¹ x x ] & Also, g(x)= ⁻¹ ( 1 2 x^2-1 ) & = ⁻¹ (2 x^2-1 )=2 ⁻¹ x & d d x g(x)= -2 1-x^2 & d f(x) d g(x) = d f(x) d x d g(x) d x & = x^ ⁻¹ x [ x 1+x^2 + ⁻¹ x x ] -2 1-x^2 & = -1 2 1-x^2 x^ ⁻¹ x [ x 1+x^2 + ⁻¹ x x ] & aligned