Partial derivatives

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0:00Choose the surface point (a,b,$f(a,b)$) on $z=f(x,y)$. To study coordinate-direction changes, hold one input fixed instead of moving both at once.

0:06Fix $y=b$. The vertical cutting plane is parallel to xz and gives the one-variable curve $z=f(x,b)$. Its tangent slope at $x=a$ is $f_x(a,b)$, the instantaneous rate when only x changes.

0:15Fix $x=a$ instead. A plane parallel to yz gives $z=f(a,y)$, whose tangent slope at $y=b$ is $f_y(a,b)$. These slopes describe different coordinate directions and need not agree.

0:26A partial derivative is an ordinary derivative with the remaining inputs fixed. Two coordinate tangents alone do not guarantee a tangent-plane approximation in all directions; differentiability requires stronger uniform error control.

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