( \left( \frac{ mg }{ k } - { L }_{ 0 } \right) \cos ( \sqrt{ \frac{ k }{ m } } t ) - { L }_{ 0 } - \frac{ mg }{ k } )
Evaluate
\frac{gm\cos(\sqrt{\frac{k}{m}}t)-L_{0}k\cos(\sqrt{\frac{k}{m}}t)-L_{0}k-gm}{k}
Differentiate w.r.t. k
\frac{\frac{L_{0}t\sin(\sqrt{\frac{k}{m}}t)}{\sqrt{\frac{k}{m}}m}-\frac{gt\sin(\sqrt{\frac{k}{m}}t)}{\sqrt{\frac{k}{m}}k}-\frac{2gm\cos(\sqrt{\frac{k}{m}}t)}{k^{2}}+\frac{2gm}{k^{2}}}{2}
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