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I have a function $z=f(x,y)$.If I take the partial derivative $f_x$ along the $x$-axis at a point $(x_0,y_0)$ then the equation of the tangent line along the $x$-axis will be

$z=z_0+f_x(x-x_0),y=y_0$

My question is this: what is the parametric form of the equation? I need an intuitive explanation.

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The simplest way is to use $x$ as a parameter: $$ \cases{ x=t&\cr y=y_0&\cr z=z_0+f_x(t-x_o) } $$

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