Higher Order Derivatives

In this tutorial we shall find the higher order derivatives. We have already seen how differentiation applied to a suitable function f\left( x \right) yields as a result another function f'\left( x \right), which assign a function of x. If f'\left( x \right) is itself differentiable, then a repetition of differentiation will result in another function that we shall denote by f''\left( x \right) and will call the second derivative of f\left( x \right) with respect to x.

Using the definition of the derivative, we have

f''\left( x \right) = \mathop {\lim }\limits_{\Delta x \to 0} \frac{{f'\left( {x + \Delta x} \right) - f'\left( x \right)}}{{\Delta x}}

Similarly, the third and the fourth derivatives are defined by

\begin{gathered} f'''\left( x \right) = \mathop {\lim }\limits_{\Delta x \to 0} \frac{{f''\left( {x + \Delta x} \right) - f''\left( x \right)}}{{\Delta x}} \\ {f^{IV}}\left( x \right) = \mathop {\lim }\limits_{\Delta x \to 0} \frac{{f'''\left( {x + \Delta x} \right) - f'''\left( x \right)}}{{\Delta x}} \\ \end{gathered}

The successive derivatives of the function y = f\left( x \right) are denoted by

y',y'',y''', \ldots ,{y^{n - 1}},{y^n}

OR

{y_1},{y_2},{y_3}, \ldots ,{y_{n - 1}},{y_n}

OR

\frac{{dy}}{{dx}},\frac{{{d^2}y}}{{d{x^2}}},\frac{{{d^3}y}}{{d{x^3}}}, \ldots ,\frac{{{d^{n - 1}}y}}{{d{x^{n - 1}}}},\frac{{{d^n}y}}{{d{x^n}}}