$f\left( x \right)=\sin \left( 4x-f\left( x \right) \right)\Rightarrow $
${f}'\left( x \right)=\left( 4-{f}'\left( x \right) \right)\cos \left( 4x-f\left( x \right) \right)\Rightarrow {f}'\left( 0 \right)=\left( 4-{f}'\left( 0 \right) \right)\cos \left( 0-f\left( 0 \right) \right)$
اما $f\left( 0 \right)=0$ ، در نتیجه:
$\Rightarrow {f}'\left( 0 \right)=\left( 4-{f}'\left( 0 \right) \right)\cos 0\Rightarrow {f}'\left( 0 \right)=4-{f}'\left( 0 \right)\Rightarrow 2{f}'\left( 0 \right)=4\Rightarrow {f}'\left( 0 \right)=2$