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.