You can compute the gradient of each band of an image with image.gradient(). For example, the following code computes the gradient magnitude and direction of the Landsat 8 panchromatic band:

// Load a Landsat 8 image and select the panchromatic band.
var image = ee.Image('LANDSAT/LC08/C01/T1/LC08_044034_20140318').select('B8');

// Compute the image gradient in the X and Y directions.
var xyGrad = image.gradient();

// Compute the magnitude of the gradient.
var gradient ='x').pow(2)

// Compute the direction of the gradient.
var direction ='y').atan2('x'));

// Display the results.
Map.setCenter(-122.054, 37.7295, 10);
Map.addLayer(direction, {min: -2, max: 2, format: 'png'}, 'direction');
Map.addLayer(gradient, {min: -7, max: 7, format: 'png'}, 'gradient');

Note that gradient() outputs two bands: the gradient in the X-direction and the gradient in the Y-direction. As shown in the example, the two directions can be combined to get gradient magnitude and direction. The magnitude should look something like Figure 1.

Figure 1. Panchromatic gradient magnitude for the Landsat 8 imagery over the San Francisco Bay area, California, USA.

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Google Earth Engine API