How would you differentiate 2lnx/x^2?

Question

How would you differentiate 2lnx/x^2?

Answer 1

d(u,v)/dx = v(du/dx) + u(dv/dx)

let v=1/x² ; u = ln(x)

The derivative of a constant times a function is the constant times the derivative of the function. i.e d/dx(af(x)) = a(d/dxf(x))

Set aside the 2 for now and focus on d/dx(ln(x)/x²)

d/dx (ln(x)/x²) = 1/x²(d/dxln(x)) + ln(x) d/dx(1/x²)

d/dx (ln(x)/x²) = 1/x²(1/x) + ln(x) (-2/x³)

d/dx (ln(x)/x²) = 1/x³ - 2ln(x)/x³

d/dx (ln(x)/x²) = (1 - 2ln(x)/x³)

2d/dx (ln(x)/x²) = d/dx (2ln(x)/x²) = 2(1 - 2ln(x)/x³)

Answer: (2 - 4 ln(x))/x³

Answer 2

f(x)== 2 Ln(x)/x²

f'(x)== 2[(1/x) x²-2x Ln(x)]/x^4 ==

== 2x[ 1-2Ln(x)]/x^4 == 2[1-2Ln(x)]/x³

Answer 3

y=2lnx/(x^2)

Quotient Rule.

y'=(2(x^2)/x-4xlnx)/(x^4)

y'=(2x-4xlnx)/(x^4)

y'=(2-4lnx)/(x^3)