Higher order derivatives and higher order differentials
      Higher order derivatives
         Higher order derivatives examples
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 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
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