If z = f ( x , y ) {\displaystyle z=f(x,y)} , then the first partial derivatives of with respect to x {\displaystyle x} and y {\displaystyle y} are the functions ∂ z ∂ x {\displaystyle {\frac {\partial z}{\partial x}}} and ∂ z ∂ x {\displaystyle {\frac {\partial z}{\partial x}}} , defined as shown. ∂ z ∂ x = lim Δ x → 0 f ( x + Δ x , y ) − f ( x , y ) Δ x {\displaystyle {\frac {\partial z}{\partial x}}=\lim _{\Delta x\to 0}{\frac {f(x+\Delta x,y)-f(x,y)}{\Delta x}}} ∂ z ∂ y = lim Δ y → 0 f ( x , y + Δ y ) − f ( x , y ) Δ y {\displaystyle {\frac {\partial z}{\partial y}}=\lim _{\Delta y\to 0}{\frac {f(x,y+\Delta y)-f(x,y)}{\Delta y}}}
Example: Find ∂ z ∂ x {\displaystyle {\frac {\partial z}{\partial x}}} and ∂ z ∂ y {\displaystyle {\frac {\partial z}{\partial y}}} of
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, then the first partial derivatives of with respect to 
and 
are the functions 
and , defined as shown.
and 
of