Correct option is A. $$44$$
Using law of exponents, $$a^{-m}=\dfrac {1}{a^m}$$
$$\Rightarrow \ (7^{-1}-8^{-1})^{-1}-(3^{-1} -4^{-1})^{-1}=\left(\dfrac {1}{7}-\dfrac {1}{8}\right)^{-1}-\left(\dfrac {1}{3}-\dfrac {1}{4}\right)^{-1}$$
$$\Rightarrow \ \left(\dfrac {1\times 8-1\times 7}{56}\right)^{-1}- \left(\dfrac {1\times 4-1\times 3}{12}\right)^{-1}$$
$$=\left(\dfrac {8-7}{56}\right)^{-1} -\left(\dfrac {4-3}{12}\right)^{-1}$$
$$=\left(\dfrac {1}{56}\right)^{-1} -\left(\dfrac {1}{12}\right)^{-1}$$
Again we will use $$a^{-m} =\dfrac {1}{a^m}$$
$$\Rightarrow (7^{-1} -8^{-1})^{-1} -(3^{-1} -4^{-1})^{-1}=\left(\dfrac {1}{56}\right)^{-1}-\left(\dfrac {1}{12}\right)^{-1}$$
$$=56-12=44$$.