Logarithmic differentiation problem?

Question

Differentiate y=4x^(x^4) with respect to x utilizing logarithmic differentiation.

Answer 1

Given that  y=4x^(x^4)

Apply log each side.

log y=log(4x^(x^4))

By using logarithm simpler formula log(a^(m)) = m log a

log y=x^(4)log(4x)

 Apply .Defferentiative  with respective x each side.

d/dx(logy)=d/dx(x^(4)log(4x))

Derivative of logarithm function d/dx(logx)=1/x

The product rule of derivatives  d/dx(uv)=ud/dx(v)+vd/dx(u)

(1/y)dy/dx=[x^(4)d/dx(log (4x)+log(4x)d/dx(x^4)]

The derivatives of constant times of function d/dx K(u(x)=Kd/dx(u(x)

(The power rule of derivatives d/dx(x^(n))=n.x^(n-1)

(1/y)dy/dx=[x^(4)(1/4x)d/dx(4x))+ (log(4x)4(x^(4-1)]

(1/y)dy/dx=[4x^(4)(1/4x)+ (log(4x)(4(x^(4-1)]

(1/y)dy/dx=[4x^(4)(1/4x)+ (log(4x)(4x^(3)

(1/y)dy/dx=[4x^(4)/4x)+ 4x^(3)(log(4x)]

(1/y)dy/dx=[x^(3))+ 4x^(3)(log(4x)]                       4x^(4)/4x]=x^4/x=x^(3)

  Take out common term (x^3) and multiply each side by 'y'

y(1/y)dy/dx=yx^(3)[1+ 4(log(4x)]

dy/dx=yx^(3)[1+ 4(log(4x)]................(1)

substitute the value of ' y' in equation (1)

dy/dx=4x^(x^4)(x^(3)[1+ 4(log(4x)]

Therefore y' = 4x^(x^4)(x^(3)[1+ 4(log(4x)].