Similarly, the partial derivative with respect to $y$ measures the slope in the direction of the $y$-axis.
, compute the dot product of the gradient and the unit vector: multivariable differential calculus
If ( F(x,y,z) = 0 ) defines ( z ) implicitly: [ \frac\partial z\partial x = -\fracF_xF_z, \quad \frac\partial z\partial y = -\fracF_yF_z ] (provided ( F_z \neq 0 )). Similarly, the partial derivative with respect to $y$
df=𝜕f𝜕xdx+𝜕f𝜕ydyd f equals partial f over partial x end-fraction d x plus partial f over partial y end-fraction d y multivariable differential calculus
The differential side of multivariable calculus primarily focuses on and linear approximations . Key topics include:
dzdt=𝜕z𝜕xdxdt+𝜕z𝜕ydydtd z over d t end-fraction equals partial z over partial x end-fraction d x over d t end-fraction plus partial z over partial y end-fraction d y over d t end-fraction Case 2: Multiple Independent Parameters has two partial derivatives: