What is the derivative of #y=(x^2lnx)^4#?
1 Answer
Aug 14, 2015
Explanation:
Notice that you can simplify your function to get
#y = (x^2)^4 * (lnx)^4#
#y = x^8 * ln^4(x)#
To differentiate this function you can use the product rule and the chain rule for
#d/dx(y) = [d/dx(x^8)] * ln^4(x) + x^8 * d/dx(ln^4(x))#
#y^' = 8x^7 * ln^4(x) + x^8 * ([d/(du)(u^4)] * d/dx(u))#
#y^' = 8x^7 * ln^4(x) + x^color(red)(cancel(color(black)(8))) * 4 ln^3(x) * 1/color(red)(cancel(color(black)(x)))#
#y^' = 8x^7 * ln^4(x) + 4x^7 * ln^3(x)#
This can be simplified to give
#y^' = color(green)(4x^7 * ln^3(x) * (2ln(x) + 1))#