Higher order derivatives and higher order differentials

Higher order derivatives
Higher order derivatives are successive derivatives of the same function. Thus, the second derivative is the derivative of the first derivative, and the derivative of the second derivative is third derivative and so on.
We denote higher order derivatives of the same function as follows,
 the second derivative is
 the third derivative is
 and the nth derivative is
Higher order derivatives examples
Example:   Find successive derivatives of   y = xn  where n Î N.
Solution:  The first derivative,    y' = n x n -1,    y'' = n (n - 1) x n - 2,    y''' = n (n - 1)(n - 2) x n - 3, ...
and the nth derivative,    y (n) = n (n - 1)(n - 2) . . . [n - (n - 1)] = 1 · 2 ·  3 · . . . · n = n!,     y (n) = n!.
Example:   Find successive derivatives of   y = sin x.
Solution:              y' = cos x = sin (x + p/2),                y'' = - sin x = sin [x + 2(p/2)],
y''' - cos x = sin [ x + 3(p/2)],      y (4) = sin x = sin [ x + 4(p/2)], ...
and the nth derivative,       y (n)  = sin [ x + n(p/2)].
Example:   Find successive derivatives of   y = cos x.
Solution:                y' = - sin x = cos (x + p/2),                y'' = - cos x = cos [x + 2(p/2)],
y''' = sin x = cos [x + 3(p/2)],             y (4) = cos x = cos [x + 4(p/2)], ...
and the nth derivative,        y (n)  = cos [x + n(p/2)].
Example:   Find successive derivatives of   y = ex.
Solution:  The first derivative,    y' = ex,     y'' = ex, ...  and the nth derivative,     y (n) = ex.
Example:   Find successive derivatives of   y = ax.
Solution:     y' = ax ln a,     y'' = ax ln2a,     y''' = ax ln3a, ...   and the nth derivative,     y (n) = ax (ln a)n.
Example:   Find successive derivatives of   y = ln (1 + x).
 Solution:
 and the nth derivative,

Calculus contents D