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