# Partial Derivative/Examples

## Examples of Partial Derivatives

### Notation for 3-Value Function

Let $u = \map f {x, y, z}$ be a real function of $3$ variables.

Then the partial derivatives may be expressed variously as:

$\dfrac {\partial u} {\partial x} = \map {f_1} {x, y, z} = \dfrac {\partial f} {\partial x} = \map {\dfrac \partial {\partial x} f} {x, y, z}$
$\dfrac {\partial u} {\partial y} = \map {f_2} {x, y, z} = \dfrac {\partial f} {\partial y} = \map {\dfrac \partial {\partial y} f} {x, y, z}$
$\dfrac {\partial u} {\partial z} = \map {f_3} {x, y, z} = \dfrac {\partial f} {\partial z} = \map {\dfrac \partial {\partial z} f} {x, y, z}$

### Arbitrary Cubic

Let $\map z {x, y}$ be the real function of $2$ variables defined as:

$z = x^3 - 3 x y + 2 y^2$

Then we have:

 $\ds \dfrac {\partial z} {\partial x}$ $=$ $\ds 3 x^2 - 3 y$ $\ds \dfrac {\partial z} {\partial y}$ $=$ $\ds -3 x + 4 y$ $\ds \dfrac {\partial^2 z} {\partial x^2}$ $=$ $\ds 6 x$ $\ds \dfrac {\partial^2 z} {\partial y^2}$ $=$ $\ds 4$ $\ds \dfrac {\partial^2 z} {\partial x \partial y} = \dfrac {\partial^2 z} {\partial y \partial x}$ $=$ $\ds -3$

### Example: $x z^y$

Let $\map f {x, y, z} = x z^y$ be a real function of $3$ variables.

Then the partial derivative with respect to the $2$nd variable may be expressed as:

$\map {f_2} {x, y, z} = x z^y \ln z$

and because of the notation chosen, we have:

$\map {f_2} {r, s, t} = r t^s \ln t$

### Example: $x^{x y}$

Let $\map f {x, y} = x^{x y}$ be a real function of $2$ variables such that $x, y \in \R_{>0}$.

Then:

 $\ds \dfrac {\partial f} {\partial x}$ $=$ $\ds x^{x y} \paren {y \ln x + y}$ $\ds \dfrac {\partial f} {\partial y}$ $=$ $\ds x^{x y + 1} \ln x$

### Example: $x \map \sin {y z}$

Let $\map f {x, y, z} = x \map \sin {y z}$ be a real function of $3$ variables.

Then:

$\map {f_3} {a, 1, \pi} = -a$

### Example: $u^2 + x^2 + y^2 = a^2$

Let $u^2 + x^2 + y^2 = a^2$ be an implicit function.

Then:

 $\ds \dfrac {\partial u} {\partial x}$ $=$ $\ds -\dfrac x u$ $\ds \dfrac {\partial u} {\partial y}$ $=$ $\ds -\dfrac y u$

### Example: $u + \ln u = x y$

Let $u + \ln u = x y$ be an implicit function.

Then:

 $\ds \dfrac {\partial u} {\partial x}$ $=$ $\ds \dfrac {u y} {u + 1}$ $\ds \dfrac {\partial u} {\partial y}$ $=$ $\ds \dfrac {u x} {u + 1}$

### Example: $v + \ln u = x y$, $u + \ln v = x - y$

Consider the simultaneous equations:

$\begin {cases} v + \ln u = x y \\ u + \ln v = x - y \end {cases}$

Then:

 $\ds \dfrac {\partial u} {\partial x}$ $=$ $\ds \dfrac {\begin {vmatrix} y u & u \\ v & 1 \end {vmatrix} } {\begin {vmatrix} 1 & u \\ v & 1 \end {vmatrix} }$ $\ds = \dfrac {u \paren {y - v} } {1 - u v}$ $\ds \dfrac {\partial v} {\partial x}$ $=$ $\ds \dfrac {\begin {vmatrix} 1 & y u \\ v & v \end {vmatrix} } {\begin {vmatrix} 1 & u \\ v & 1 \end {vmatrix} }$ $\ds = \dfrac {v \paren {1 - y u} } {1 - u v}$

### Example: $u - v + 2 w = x + 2 z$, $2 u + v + 2 w = 2 x - 2 z$, $u - v + w = z = y$

Let:

 $\ds u - v + 2 w$ $=$ $\ds x + 2 z$ $\ds 2 u + v - 2 w$ $=$ $\ds 2 x - 2 z$ $\ds u - v + w$ $=$ $\ds z - y$

Then:

 $\ds \dfrac {\partial u} {\partial y}$ $=$ $\ds 0$ $\ds \dfrac {\partial v} {\partial y}$ $=$ $\ds 2$ $\ds \dfrac {\partial w} {\partial y}$ $=$ $\ds 1$

### Example: $2 u + 3 v = \sin x$, $u + 2 v = x \cos y$

Consider the simultaneous equations:

$\begin {cases} 2 u + 3 v & = \sin x \\ u + 2 v & = x \cos y \end {cases}$

Then:

$\map {u_1} {\dfrac \pi 2, \pi} = 3$

### Example: $u^2 + v^2 = x^2$, $2 u v = 2 x y + y^2$

Consider the simultaneous equations:

 $\ds u^2 + v^2$ $=$ $\ds x^2$ $\ds 2 u v$ $=$ $\ds 2 x y + y^2$

Then:

$\map {u_1} {1, -2} = 1$

at $u = 1$, $v = 0$.