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The curly d symbol ($\partial$) represents a partial derivative. It means taking the derivative of a multivariable function with respect to one specific variable at a time, while treating all other variables as constants.

Conditions: The function has more than one independent variable.; All variables except the differentiation variable are treated as constants during the partial derivative.

The curly d symbol ($\partial$) represents a partial derivative. It means taking the derivative of a multivariable function with respect to one specific variable at a time, while treating all other variables as constants.

Conditions: The function has more than one independent variable.; All variables except the differentiation variable are treated as constants during the partial derivative.

On a two-variable surface, partial derivatives represent the slopes of the tangent plane in the coordinate directions. The partial derivative with respect to $x$ is the slope along the $x$-direction, and the partial derivative with respect to $y$ is the slope along the $y$-direction.

Conditions: The surface is locally smooth enough to have a tangent plane.; The variables are distinguished as $x$ and $y$.

On a two-variable surface, partial derivatives represent the slopes of the tangent plane in the coordinate directions. The partial derivative with respect to $x$ is the slope along the $x$-direction, and the partial derivative with respect to $y$ is the slope along the $y$-direction.

Conditions: The surface is locally smooth enough to have a tangent plane.; The variables are distinguished as $x$ and $y$.