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We present results from monitoring observations of the gravitationally lensed quasar RX J1131-1231 performed with the Chandra X-Ray Observatory. The X-ray observations were planned with relatively long exposures that allowed a search for energy-dependent microlensing in the soft (0.2-2 keV) and hard (2-10 keV) light curves of the images of RX J1131-1231. We detect significant microlensing in the X-ray light curves of images A and D, and energy-dependent microlensing of image D. The magnification of the soft band appears to be larger than that in the hard band by a factor of ∼1.3 when image D becomes more magnified. This can be explained by the difference between a compact, softer-spectrum corona that is producing a more extended, harder spectrum reflection component off the disk. This is supported by the evolution of the fluorescent iron line in image D over three consecutive time-averaged phases of the light curve. In the first period, an Fe line at E = 6.35^+0.14 _- 0.14_ keV is detected (at >99% confidence). In the second period, two Fe lines are detected, one at E = 5.50^+0.03_- 0.08_ keV (detected at >99% confidence) and another at E = 6.04^+0.10_- 0.07_ keV (marginally detected at >90% confidence), and in the third period, a broadened Fe line at 6.42^+0.16 _- 0.14_ keV is detected (at >99% confidence). This evolution of the Fe line profile during the microlensing event is consistent with the line distortion expected when a caustic passes over the inner disk where the shape of the fluorescent Fe line is distorted by general relativistic and Doppler effects.