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'd 在这里是had的缩写。had better 是个固定词组，性质为副词，意思是：最好。had better + 动词原形，即 had better do sth.

start working hard 在这里也可以写成 start to work hard，在这里用进行时更有马上开始努力工作的意思。

You'd better call him。 对的，“你最好打电话给他。”

d是had

start doing sth, 固定用法。

you'd better call him， 完全正确。

'd 是had的缩写

had better do something
应该作某事,最好还是作某事。这是固定用法。

无论什么时态，都是固定的。

You'd better = you WOULD better = 你最好...

"You'd better start working hard."
Or "You'd better start TO work hard"

"You'd better call him" is correct. 是正确的。