Examples of Higher Order Derivatives

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

We have the given function

Differentiating both sides with respect to $x$, we have

Again, differentiating both sides with respect to $x$, we have

Example: If $y = a\cos x + b\sin x$, then show that

We have the given function

Differentiating both sides with respect to $x$, we have

Again, differentiating both sides with respect to $x$, we have

Using the value of the given function $y = a\cos x + b\sin x$ in the above equation, we have