Examples of Higher Order Derivatives

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

We have the given function

y = \cos \left( {ax + b} \right)

Differentiate 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 differentiate 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

Differentiate 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 differentiate 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 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}

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