Question

Differentiate the Logarithmic functions:

𝑦 = 𝑥^3 𝑙𝑛^2 𝑥 + 𝑥^2 𝑙𝑛𝑥^3 + 1

Answer 1

y’ = 3x² ln²x + x² + 2x. 𝑙𝑛𝑥³ + 3x

𝑦 = 𝑥³ 𝑙𝑛²x+ 𝑥² 𝑙𝑛𝑥³ + 1

y = x³ ln² x+ x² (3lnx) + 1

let x³ ln² x= u and ln² x= p, x³= q

then, u’ = pq’ + qp’

q’ = 3x²

from p = ln² x

let lnx = u

u'= 1/x

then p = u²

p' = 2u

therefore, p' = 2u. 1/x

p' = 2/x lnx

u’ = ln² x. 3x² + x³ . 2/x lnx

u’ = 3x² ln² x+ 2x² lnx

y’ = 3x² ln² x+2x² lnx + 2x. 𝑙𝑛𝑥³ + X² (3/x)

y’ = 3x² ln² x+ 2x² lnx+ 2x. 𝑙𝑛𝑥³ + 3x