Differentiation of large function

Question

If !!y=a^(x^(a^(x^(.....oo)))!! . Then !!dy/dx !! at !!(x=1 and y=2)!!

(a) log 2

(b) log 16

(c) log 64

(d) None

Solution

!!y=a^(x^(a^(x^(.....oo))).....(1)!!

Take log both the sides,

!!logy = x^(a^(x^(.....oo)) loga!!

From equation !!(1)!!, we get

!!logy = x^y loga ...(2)!!

Differentiate with respect to !!x!!,

!!1/y dy/dx = x^y[y/x+logx dy/dx]loga!!

Putting the value !!x=1!! and !!y=2!!, we get

!! 1/2 dy/dx = (1)^2[2/1+log1 dy/dx] loga!!

!!=> dy/dx = 4loga....(3)!!

From equation !!(2)!!, we get

!!log2 = 1. loga=> loga = log2!!

Put the value of !!loga!! in the equation !!(3)!!

!!=> dy/dx = 4log2= log(2^4)=log16!!

Or

You can take log twice.

Taking log twice both the sides, we get

!!log(logy)= y logx + log(loga)...(4)!!

In this method you don't have to find the value of loga. Differentiate the equation !!(4)!! with respect to !!x!! and solve it.
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