To find
[tex]\frac{d^{114}}{dx^{114}}(sinx)[/tex]
We need to find some few derivatives of [tex]sinx[/tex], say the first eight derivatives and then observe some pattern.
[tex]\frac{dy}{dx}(sinx)=cosx[/tex]
[tex]\frac{d^{2}}{dx^{2}}(sinx)=-sinx[/tex]
[tex]\frac{d^{3}}{dx^{3}}(sinx)=-cosx[/tex]
[tex]\frac{d^{4}}{dx^{4}}(sinx)=sinx[/tex]
[tex]\frac{d^{5}}{dx^{5}}(sinx)=cosx[/tex]
[tex]\frac{d^{6}}{dx^{6}}(sinx)=-sinx[/tex]
[tex]\frac{d^{7}}{dx^{7}}(sinx)=-cosx[/tex]
[tex]\frac{d^{8}}{dx^{8}}(sinx)=sinx[/tex]
We can recognize the following patterns in the order of the derivativatives;
1. The derivative of [tex]sinx[/tex] to the orders, [tex]1,5,9,13,...,4n-3[/tex] is [tex]cosx[/tex]
2. The derivative of [tex]sinx[/tex] to the orders, [tex]2,6,10,14,...,4n-2[/tex] is [tex]-sinx[/tex]
3. The derivative of [tex]sinx[/tex] to the orders, [tex]3,7,11,15,...,4n-1[/tex] is [tex]-cosx[/tex]
The order, 114 belongs to the second pattern. Therefore,
[tex]\frac{d^{114}}{dx^{114}}(sinx)=-sinx[/tex]
