Let f:(- , )- 0 R be a differentiable function such that f^ (1)= _ a a^2 f ( 1 a ) . Then _ a a(a+1) 2 ⁻¹ ( 1 a )+a^2-2 _e a is equal to

Step-by-step solution

aligned & f:(- , )- 0 R & f^ (1)= _ a a^2 f ( 1 a ) & _ a a(a+1) 2 ⁻¹ ( 1 a )+a^2-2 (a) & _ a a^2 ( (1+ 1 a ) 2 ⁻¹ ( 1 a )+1- 2 a^2 (a) ) & f(x)= 1 2 (1+x) ⁻¹(x)+1-2 x^2 (x) aligned aligned & f ^ ( x )= 1 2 ( 1+ x 1+ x ^2 + ⁻¹( x )+4 x ( x ) )+2 x & f ^ (1)= 1 2 (1+ 4 )+2 & f ^ (1)= 5 2 + 8 aligned