# 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
$y’,y”,y”’, \ldots ,{y^{n – 1}},{y^n}$

OR
${y_1},{y_2},{y_3}, \ldots ,{y_{n – 1}},{y_n}$

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