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Different functions such as $x^2$ and $x^2 + C$ share the same derivative because the derivative measures the slope of the tangent line at each point $x$. Adding a constant $C$ shifts the graph vertically but does not change its steepness or shape.

Conditions: The functions differ only by an additive constant.; The domain is an interval where the derivative exists.

Different functions such as $x^2$ and $x^2 + C$ share the same derivative because the derivative measures the slope of the tangent line at each point $x$. Adding a constant $C$ shifts the graph vertically but does not change its steepness or shape.

Conditions: The functions differ only by an additive constant.; The domain is an interval where the derivative exists.