Cytoskeletal filaments provide cells with mechanical stability and organization. The main key players are actin filaments and microtubules governing a cell's response to mechanical stimuli. We investigated the specific influences of these crucial components by deforming MCF-7 epithelial cells at small (5% deformation) and large strains (>5% deformation). To understand specific contributions of actin filaments and microtubules, we systematically studied cellular responses after treatment with cytoskeleton influencing drugs. Quantification with the microfluidic optical stretcher allowed capturing the relative deformation and relaxation of cells under different conditions. We separated distinctive deformational and relaxational contributions to cell mechanics for actin and microtubule networks for two orders of magnitude of drug dosages. Disrupting actin filaments via latrunculin A, for instance, revealed a strain-independent softening. Stabilizing these filaments by treatment with jasplakinolide yielded cell softening for small strains but showed no significant change at large strains. In contrast, cells treated with nocodazole to disrupt microtubules displayed a softening at large strains but remained unchanged at small strains. Stabilizing microtubules within the cells via paclitaxel revealed no significant changes for deformations at small strains, but concentration-dependent impact at large strains. This suggests that for suspended cells, the actin cortex is probed at small strains, while at larger strains; the whole cell is probed with a significant contribution from the microtubules.