Comment on Activity 4

By the Chain Rule, the derivative of ${\displaystyle sin\text{(}\omega t\text{)}}$ is ${\displaystyle \omega cos\text{(}\omega t\text{)}}$. When we differentiate again, using both the Chain Rule and the Constant Multiple Rule, we find that the second derivative of ${\displaystyle sin\text{(}\omega t\text{)}}$ is ${\displaystyle -{\omega }^{2}sin\text{(}\omega t\text{)}}$, as asserted in part (a). The calculation for ${\displaystyle cos\text{(}\omega t\text{)}}$ is similar.

These calculations show that the effect of differentiating either ${\displaystyle sin\text{(}\omega t\text{)}}$ or ${\displaystyle cos\text{(}\omega t\text{)}}$ twice is to multiply the original function by ${\displaystyle -{\omega }^2.}$ That is, each of these functions satisfies the differential equation in part (b).

For part (c), we can combine these results in another calculation using the Sum and Constant Multiple Rules: