How can I find derivative of this function

Question

if y=5^6^7^x

how can I find the derivative of this function by using the chain rule?

I want to know the process and the answer as soon as possible.

Answer 1

Let u = 7^x then the problem becomes:

y = 5^6^u

Now let w = 6^u then the problem becomes:

y = 5^w

Take the natural log of both sides:

ln(y) = ln(5^w)

ln(y) = w(ln(5))

Take the derivative of both sides:

(1/y)•(y') = ln(5)•w' but w is a complex function so we need to work that separately:

Since w = 6^u

ln(w) = ln(6^u)

ln(w) = u(ln(6))

Taking the derivative:

(1/w)•w' = ln(6)•u' but u is a complex function so we need to work that separately:

u = 7^x

Take the natural log of both sides to get:

ln(u) = ln(7^x)

ln(u) = x(ln(7))

Take the derivative of both sides:

(1/u)•u' = ln(7)

u' = u(ln(7)) but u = 7^x so:

u' = (7^x)(ln(7))

Using our (1/w)•w' = ln(6)•u' and plugging in u' we get:

(1/w)•w' = ln(6)•(7^x)(ln(7)) meaning:

w' = ln(6)•(7^x)(ln(7))•w but w = 6^u = 6^7^x so:

w' = ln(6)•(7^x)(ln(7))•(6^7^x)

Using our (1/y)•(y') = ln(5)•w' and plugging in the w':

(1/y)•(y') = ln(5)•ln(6)•(7^x)(ln(7))•(6^7^x)

y' = [ln(5)•ln(6)•(7^x)(ln(7))•(6^7^x)]•y but y = 5^6^7^x so:

y' = [ln(5)•ln(6)•(7^x)(ln(7))•(6^7^x)]•(5^6^7^x)

So, reordering this to make it "prettier":

y' = ln(5)ln(6)ln(7)(7^x)(6^7^x)(5^6^7^x)

Answer 2