Question #b61f5

1 Answer
Feb 6, 2017

G'(2)=-1.

Explanation:

Knowing that, g is the inverse function of f, we have,

g@f"=the Identity fun. "rArr g(f(x))=I(x)=x....(star)

rArr d/dx g(f(x))=d/dx(x).

"By the Chain Rule, "g'(f(x))f'(x)=1.

In particular, for x=3, g'(f(3))f'(3)=1................(ast)

But, f(3)=2, and, f'(3)=1/9 :. (ast) rArr g'(2)=9..................(1)

Now, G(x)=1/g(x)rArr G'(x)=-(g'(x))/[g(x)]^2..."[the Chain Rule]"

Hence, for x=2, G'(2)=-(g'(2))/[g(2)]^2=-9/[g(2)]^2, ........[because, (1)]

"Here, to find, "g(2)," we take "x=3" in "(star)" & get, "g(f(3))=3, i.e., g(2)=3," as "f(3)=2.

So, finally, G'(2)=-9/3^2=-1.

Enjoy Maths., and, Spread the Joy!