Answer
$F = \frac{m_0~a}{(1-\frac{v^2}{c^2})^{3/2}}$
Work Step by Step
$F = (\frac{d}{dt})(m~v)$
$F = (\frac{d}{dt})(\frac{m_0~v}{\sqrt{1-\frac{v^2}{c^2}}})$
$F = \frac{m_0~\frac{dv}{dt}~\sqrt{1-\frac{v^2}{c^2}}-\frac{1}{2}(1-\frac{v^2}{c^2})^{-1/2}~(-\frac{2v}{c^2})(\frac{dv}{dt})~(m_0~v)}{1-\frac{v^2}{c^2}}$
$F = m_0~\frac{dv}{dt}\cdot ~\frac{\sqrt{1-\frac{v^2}{c^2}}+(1-\frac{v^2}{c^2})^{-1/2}~(\frac{v^2}{c^2})}{1-\frac{v^2}{c^2}}$
$F = m_0~\frac{dv}{dt}\cdot ~\frac{\sqrt{1-\frac{v^2}{c^2}}+(1-\frac{v^2}{c^2})^{-1/2}~(\frac{v^2}{c^2})}{1-\frac{v^2}{c^2}}\cdot \frac{\sqrt{1-\frac{v^2}{c^2}}}{\sqrt{1-\frac{v^2}{c^2}}}$
$F = m_0~a\cdot ~\frac{(1-\frac{v^2}{c^2})+\frac{v^2}{c^2}}{(1-\frac{v^2}{c^2})^{3/2}}$
$F = m_0~a\cdot ~\frac{1}{(1-\frac{v^2}{c^2})^{3/2}}$
$F = \frac{m_0~a}{(1-\frac{v^2}{c^2})^{3/2}}$
