The simplest explanation is that the HTML rendered in your code does not match what you’re actually looking for. Without seeing the JS responsible for the slider, it’s impossible to say where you went wrong, but you should compare the mocked-up version (that works) with what you’re actually getting (HTML source after page render) to see if there were any typos or other potential mistakes.
Another possibility is that, although the URL to your image is correct, it has some strange character in there. Since you’re doing this within an HTML attribute, I would suggest running your echo statement through sanitization using esc_attr():
<a href="#" title="Dum Slider 1" target="_blank"><img src="<?php echo esc_attr($slide_img_src); ?>" class="max-image" width="" height="" alt="Slide 1"></a>
If possible, would you mind sharing the slider code?