By the Chain Rule, the derivative of `sintext[(]omega t text[)]` is `omega costext[(]omega t text[)]`. When we differentiate again, using both the Chain Rule and the Constant Multiple Rule, we find that the second derivative of `sintext[(]omega t text[)]` is `-omega^2 sintext[(]omega t text[)]`, as asserted in part (a). The calculation for `costext[(]omega t text[)]` is similar.
These calculations show that the effect of differentiating either `sintext[(]omega t text[)]` or `costext[(]omega t text[)]` twice is to multiply the original function by `-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:
| `(d^2x)/(dt^2)` | `=d^2/(dt^2)[A sintext[(]omega t text[)]+B costext[(]omega t text[)]]` |
| `=A d^2/(dt^2)sintext[(]omega t text[)]+B d^2/(dt^2) costext[(]omega t text[)]` | |
| `=A [-omega^2 sintext[(]omega t text[)]]+B [-omega^2 costext[(]omega t text[)]]` | |
| `=-omega^2[A sintext[(]omega t text[)]+B costext[(]omega t text[)]]` | |
| `=-omega^2x.` |