Differentiation (5 stars)

Find the second derivative of y = ln x / x^2.

Find a FORMULA for the second derivative of y = x^n (ln x)

Please show the way...thanks.

y = ln(x) / x^2
y = x^(-2) * ln(x)

u = x^(-2)
du = -2 * x^(-3)
v = ln(x)
dv = x^(-1)


y' = u * dv + v * du
y' = x^(-2) * x^(-1) - 2 * x^(-3) * ln(x)
y' = (1 - 2 * ln(x)) / x^3


u = 1 - 2 * ln(x)
du = -2/x
v = x^3
dv = 3x^2

(v * du - u * dv) / v^2 =>
(x^3 * (-2/x) - (1 - 2 * ln(x)) * 3x^2) / x^6 =>
(-2x^2 - 3x^2 + 6x^2 * ln(x)) / x^6 =>
(-5 + 6 * ln(x)) / x^4 =>
(6 * ln(x) - 5) / x^4




y = x^n * ln(x)
u = x^n
du = n * x^(n - 1)
v = ln(x)
dv = 1/x

y' = u * dv + v * du
y' = x^n * (1/x) + n * x^(n - 1) * ln(x)
y' = x^(n - 1) + n * x^(n - 1) * ln(x)
y' = x^(n - 1) * (1 + n * ln(x))

u = x^(n - 1)
du = (n - 1) * x^(n - 2)
v = 1 + n * ln(x)
dv = n/x

y'' = x^(n - 1) * n / x + (n - 1) * x^(n - 2) * (1 + n * ln(x))
y'' = x^(n - 2) * (n + (n - 1) * (1 + n * ln(x)))