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The notation $\frac{d}{dx}f(-2)$ represents evaluating the derivative of the function $f$ specifically at the horizontal input coordinate $x = -2$. It denotes the instantaneous slope of the tangent line to the graph at that exact abscissa, rather than referring to a point where the output or function value happens to be $-2$.

Conditions: The expression evaluates the derivative operator applied to the function $f$.; The argument inside the parentheses is strictly the independent variable (input $x$).

The notation $\frac{d}{dx}f(-2)$ represents evaluating the derivative of the function $f$ specifically at the horizontal input coordinate $x = -2$. It denotes the instantaneous slope of the tangent line to the graph at that exact abscissa, rather than referring to a point where the output or function value happens to be $-2$.

Conditions: The expression evaluates the derivative operator applied to the function $f$.; The argument inside the parentheses is strictly the independent variable (input $x$).