อนุพันธ์ของ ( 4 x − 2 ) / ( x 2 + 1 ) {\displaystyle (4x-2)/(x^{2}+1)} คือ:

d d x ( 4 x − 2 ) x 2 + 1 {\displaystyle {\frac {d}{dx}}{\frac {(4x-2)}{x^{2}+1}}}	= ( x 2 + 1 ) ( 4 ) − ( 4 x − 2 ) ( 2 x ) ( x 2 + 1 ) 2 {\displaystyle ={\frac {(x^{2}+1)(4)-(4x-2)(2x)}{(x^{2}+1)^{2}}}}
	= ( 4 x 2 + 4 ) − ( 8 x 2 − 4 x ) ( x 2 + 1 ) 2 {\displaystyle ={\frac {(4x^{2}+4)-(8x^{2}-4x)}{(x^{2}+1)^{2}}}}
	= − 4 x 2 + 4 x + 4 ( x 2 + 1 ) 2 {\displaystyle ={\frac {-4x^{2}+4x+4}{(x^{2}+1)^{2}}}}

อนุพันธ์ของ sin ⁡ ( x ) / x 2 {\displaystyle \sin(x)/x^{2}} (เมื่อ x {\displaystyle x} ≠ 0) คือ:

cos ⁡ ( x ) x 2 − sin ⁡ ( x ) 2 x x 4 {\displaystyle {\frac {\cos(x)x^{2}-\sin(x)2x}{x^{4}}}}