Paragraph: Let f:(-1,1) R be a differentiable function with f(0)=-1 and f^ (0)=1 . Let h(x)=f(2 f(x)+2) and g(x)=(h(x))^2 . Question: What is g^ (0) equal to?

Step-by-step solution

Let f :(-1,1) R be a differentiable function with f (0)=-1 and f^ (0)=1 g ( n )=( h ( x ))^2 Differentiate both side w.r.t x g ^ ( x )=2( ~h ( x )) h ^ ( x ) aligned & g^ (0)=2(h(0)) h^ (0) & [ h^ (0)=2 ] & [ f^ (0)=-1 ] & =2 f(2+10)+2) 2 & =2 f(2(-1)+2)=2 f(0) 2=(-2) 2=-4 . aligned