Partial Derivatives

A partial derivative is a Derivative taken with respect to just one variable, holding all other variables constant.

For example, let $f(x,y)=y\sin (x^2+y^3)$. Then the partial derivative with respect to $x$ is:

$$\frac{\partial f}{\partial x}=\frac{\partial }{\partial x}\left(y\sin (x^2+y^3)\right)=y\cos (x^2+y^3)\cdot 2x=2xy\cos (x^2+y^3)$$

and the partial derivative with respect to $y$ is:

$$\frac{\partial f}{\partial y}=\frac{\partial }{\partial y}\left(y\sin (x^2+y^3)\right)=\sin (x^2+y^3)+y\cos (x^2+y^3)\cdot 3y^2=\sin (x^2+y^3)+3y^3\cos (x^2+y^3)$$

Link:

• Jacobian