# Examples of Higher Order Derivatives

Example: Find the second derivative ${y_2}$ if $y = \cos \left( {ax + b} \right)$.

We have the given function $y = \cos \left( {ax + b} \right)$

Differentiating both sides with respect to $x$, we have
$\begin{gathered} \frac{{dy}}{{dx}} = \frac{d}{{dx}}\cos \left( {ax + b} \right) \\ \Rightarrow {y_1} = – \sin \left( {ax + b} \right)\frac{d}{{dx}}\left( {ax + b} \right) \\ \Rightarrow {y_1} = – a\sin \left( {ax + b} \right) \\ \end{gathered}$

Again, differentiating both sides with respect to $x$, we have
$\begin{gathered} \frac{{d{y_1}}}{{dx}} = – a\frac{d}{{dx}}\sin \left( {ax + b} \right) \\ \Rightarrow {y_2} = – a\cos \left( {ax + b} \right)\frac{d}{{dx}}\left( {ax + b} \right) \\ \Rightarrow {y_2} = – {a^2}\cos \left( {ax + b} \right) \\ \end{gathered}$

Example: If $y = a\cos x + b\sin x$, then show that $\frac{{{d^2}y}}{{d{x^2}}} + y = 0$

We have the given function
$y = a\cos x + b\sin x$

Differentiating both sides with respect to $x$, we have
$\begin{gathered} \frac{{dy}}{{dx}} = \frac{d}{{dx}}\left( {a\cos x + b\sin x} \right) \\ \Rightarrow \frac{{dy}}{{dx}} = a\frac{d}{{dx}}\cos x + b\frac{d}{{dx}}\sin x \\ \Rightarrow \frac{{dy}}{{dx}} = – a\sin x + b\cos x \\ \end{gathered}$

Again, differentiating both sides with respect to $x$, we have
$\begin{gathered} \frac{{{d^2}y}}{{d{x^2}}} = – a\frac{d}{{dx}}\sin x + b\frac{d}{{dx}}\cos x \\ \Rightarrow \frac{{{d^2}y}}{{d{x^2}}} = – a\cos x – b\sin x \\ \Rightarrow \frac{{{d^2}y}}{{d{x^2}}} = – \left( {a\cos x + b\sin x} \right) \\ \end{gathered}$

Using the value of the given function $y = a\cos x + b\sin x$ in the above equation, we have
$\begin{gathered} \frac{{{d^2}y}}{{d{x^2}}} = – y \\ \Rightarrow \frac{{{d^2}y}}{{d{x^2}}} + y = 0 \\ \end{gathered}$