A "sharp" corner is one where the slope of the function changes values non-continuously from one value to another. The value of the slope of the function, plotted on the same x axis, is the derivative, and at the point where the corner occurs it jumps from one value to a totally different value. At that point it has no definite value; it is said to not exist at that point.
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If you would calculate them in the normal way y' = Δy/Δx with Δ going to zero, you would get different values wether you are at the left, the middle or the right of the sharp corner. You wouldn't know which one of the three to choose, therefore we say that there is no value.
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it doesn't exist because a derivative is the slope of a tangent line. At a sharp corner, there is no tangent line, because there is no defined slope at the sharp corner.