Higher Order Derivatives

In this tutorial we shall find the higher order derivatives. We have already seen how differentiation is applied to a suitable function $$f\left( x \right)$$ and results in another function $$f’\left( x \right)$$. 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
\[\begin{gathered} f”’\left( x \right) = \mathop {\lim }\limits_{\Delta x \to 0} \frac{{f”\left( {x + \Delta x} \right) – f”\left( x \right)}}{{\Delta x}} \\ {f^{IV}}\left( x \right) = \mathop {\lim }\limits_{\Delta x \to 0} \frac{{f”’\left( {x + \Delta x} \right) – f”’\left( x \right)}}{{\Delta x}} \\ \end{gathered} \]

The successive derivatives of the function $$y = f\left( x \right)$$ are denoted by
\[y’,y”,y”’, \ldots ,{y^{n – 1}},{y^n}\]

\[{y_1},{y_2},{y_3}, \ldots ,{y_{n – 1}},{y_n}\]

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