The derivative of \(\sin(x)\) is \(\cos(x)\) and the derivative of \(\cos(x)\) is \(-\sin(x).\) Altogether, \[ \frac{\mathrm{d}}{\mathrm{d}x}\biggl( \sin(x) \biggr) = \cos(x) \qquad\text{ and }\qquad \frac{\mathrm{d}}{\mathrm{d}x}\biggl( \cos(x) \biggr) = -\sin(x) \] Because of that negative sign, both the sine and cosine functions are their own fourth derivatives.