Math1231 Partial Derivatives

A Partial Derivative is a multivariable function which for differentiation has all variables aside from your dF/d? as constant, and the differentiation is done in regards to ?

e.g.

(1)

\begin{align} \begin {array} {l l} F(x,y,z) & = x^3yz + 5z \\ \frac {\partial{F}}{\partial{x}} & = 3x^2yz \\ \frac {\partial{F}}{\partial{y}} & = x^3z \\ \frac {\partial{F}}{\partial{z}} & = x^3y + 5 \\ \end {array} \end{align}

Second Partial Derivatives

With two variables there are 4 formats we can do this in.

(2)

\begin{align} \frac {\partial^2F}{\partial x^2} = \frac {\partial}{\partial x} \frac{\partial F}{\partial x} \\ \frac {\partial^2F}{\partial y^2} = \frac {\partial}{\partial y} \frac{\partial F}{\partial y} \\ \frac {\partial^2F}{\partial x\partial y} = \frac {\partial}{\partial x} \frac{\partial F}{\partial y} \\ \frac {\partial^2F}{\partial y\partial x} = \frac {\partial}{\partial y} \frac{\partial F}{\partial x} \\ \end{align}

e.g.

(3)

\begin{align} \begin {array} {r l} F(x,y) & = e^x sin(y) \\ \\ \frac {\partial F}{\partial x} & = e^xsin(y) \\ \frac {\partial F}{\partial y} & = e^xcos(y) \\ \\ \frac {\partial^2 F}{\partial x^2} & = e^xsin(y) \\ \frac {\partial^2 F}{\partial y^2} & = -e^xsin(y) \\ \frac {\partial^2 F}{\partial x\partial y} & = e^xcos(y) \\ \frac {\partial^2 F}{\partial y\partial x} & = e^xcos(y) \\ \end {array} \end{align}

See: Bernoulli's Theorem

Chain Rules for Partial Derivatives