Let X, Y be complex Banach spaces. Let {mathcal L}(X,Y) be the space of bounded operators. An important aspect of understanding differentiability is to study the subdifferential of the norm at a point, say x in X, this is the set, {f in X^*:Vert fVert =1~,f(x)=Vert xVert }. See page 7 in Deville et al. (Pitman Monographs and Surveys in Pure and Applied Mathematics. 64. Harlow: Longman Scientific and Technical. New York: John Wiley and Sons, Inc. 1993). Motivated by recent results of Singla (Singla in Linear Alg. Appl. 629:208–218, 2021) in the context of Hilbert spaces, for T in {{mathcal {L}}}(X,Y), we determine the subdifferential of the operator norm at T, partial _T = {Lambda in {{mathcal {L}}}(X,Y)^*: Lambda (T) = Vert TVert ~,~Vert Lambda Vert =1}. Our approach is based on the ‘position’ of the space of compact operators and the Calkin norm of T. Our ideas give a unified approach and extend several results from Singla (Linear Alg. Appl. 629:208–218, 2021) to the case of ell ^p-spaces for 1<p<infty . We also investigate the converse, using the structure of the subdifferential set to decide when the Calkin norm is a strict contraction. As an application of these ideas, we partially solve the open problem of relating the subdifferential of the operator norm at a compact operator T to that of T(x_0), where x_0 is a unit vector where T attains its norm.