4.3 Partial derivatives (Page 4/11)

Calculus volume 3 Page 4 / 11

We can calculate a partial derivative of a function of three variables using the same idea we used for a function of two variables. For example, if we have a function $f$ of $x, y, and z,$ and we wish to calculate $\partial f / \partial x,$ then we treat the other two independent variables as if they are constants, then differentiate with respect to $x .$

Calculating partial derivatives for a function of three variables

Use the limit definition of partial derivatives to calculate $\partial f / \partial x$ for the function

f (x, y, z) = x^{2} - 3 x y + 2 y^{2} - 4 x z + 5 y z^{2} - 12 x + 4 y - 3 z .

Then, find $\partial f / \partial y$ and $\partial f / \partial z$ by setting the other two variables constant and differentiating accordingly.

We first calculate $\partial f / \partial x$ using [link] , then we calculate the other two partial derivatives by holding the remaining variables constant. To use the equation to find $\partial f / \partial x,$ we first need to calculate $f (x + h, y, z) :$

\begin{matrix} f (x + h, y, z) & = {(x + h)}^{2} - 3 (x + h) y + 2 y^{2} - 4 (x + h) z + 5 y z^{2} - 12 (x + h) + 4 y - 3 z \\ = x^{2} + 2 x h + h^{2} - 3 x y - 3 x h + 2 y^{2} - 4 x z - 4 h z + 5 y z^{2} - 12 x - 12 h + 4 y - 3 z \end{matrix}

and recall that $f (x, y, z) = x^{2} - 3 x y + 2 y^{2} - 4 z x + 5 y z^{2} - 12 x + 4 y - 3 z .$ Next, we substitute these two expressions into the equation:

\begin{matrix} \frac{\partial f}{\partial x} & = lim_{h \to 0} [\frac{x^{2} + 2 x h + h^{2} - 3 x y - 3 h y + 2 y^{2} - 4 x z - 4 h z + 5 y z^{2} - 12 x - 12 h + 4 y - 3 z}{h} \\ - \frac{x^{2} - 3 x y + 2 y^{2} - 4 x z + 5 y z^{2} - 12 x + 4 y - 3 z}{h}] \\ = lim_{h \to 0} [\frac{2 x h + h^{2} - 3 h y - 4 h z - 12 h}{h}] \\ = lim_{h \to 0} [\frac{h (2 x + h - 3 y - 4 z - 12)}{h}] \\ = lim_{h \to 0} (2 x + h - 3 y - 4 z - 12) \\ = 2 x - 3 y - 4 z - 12. \end{matrix}

Then we find $\partial f / \partial y$ by holding $x and z$ constant. Therefore, any term that does not include the variable $y$ is constant, and its derivative is zero. We can apply the sum, difference, and power rules for functions of one variable:

\begin{matrix} \frac{\partial}{\partial y} [x^{2} - 3 x y + 2 y^{2} - 4 x z + 5 y z^{2} - 12 x + 4 y - 3 z] \\ = \frac{\partial}{\partial y} [x^{2}] - \frac{\partial}{\partial y} [3 x y] + \frac{\partial}{\partial y} [2 y^{2}] - \frac{\partial}{\partial y} [4 x z] + \frac{\partial}{\partial y} [5 y z^{2}] - \frac{\partial}{\partial y} [12 x] + \frac{\partial}{\partial y} [4 y] - \frac{\partial}{\partial y} [3 z] \\ = 0 - 3 x + 4 y - 0 + 5 z^{2} - 0 + 4 - 0 \\ = −3 x + 4 y + 5 z^{2} + 4. \end{matrix}

To calculate $\partial f / \partial z,$ we hold x and y constant and apply the sum, difference, and power rules for functions of one variable:

\begin{matrix} \frac{\partial}{\partial z} [x^{2} - 3 x y + 2 y^{2} - 4 x z + 5 y z^{2} - 12 x + 4 y - 3 z] \\ = \frac{\partial}{\partial z} [x^{2}] - \frac{\partial}{\partial z} [3 x y] + \frac{\partial}{\partial z} [2 y^{2}] - \frac{\partial}{\partial z} [4 x z] + \frac{\partial}{\partial z} [5 y z^{2}] - \frac{\partial}{\partial z} [12 x] + \frac{\partial}{\partial z} [4 y] - \frac{\partial}{\partial z} [3 z] \\ = 0 - 0 + 0 - 4 x + 10 y z - 0 + 0 - 3 \\ = −4 x + 10 y z - 3. \end{matrix}

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Use the limit definition of partial derivatives to calculate $\partial f / \partial x$ for the function

f (x, y, z) = 2 x^{2} - 4 x^{2} y + 2 y^{2} + 5 x z^{2} - 6 x + 3 z - 8 .

Then find $\partial f / \partial y$ and $\partial f / \partial z$ by setting the other two variables constant and differentiating accordingly.

$\frac{\partial f}{\partial x} = 4 x - 8 x y + 5 z^{2} - 6, \frac{\partial f}{\partial y} = −4 x^{2} + 4 y, \frac{\partial f}{\partial z} = 10 x z + 3$

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Calculating partial derivatives for a function of three variables

Calculate the three partial derivatives of the following functions.

$f (x, y, z) = \frac{x^{2} y - 4 x z + y^{2}}{x - 3 y z}$
$g (x, y, z) = sin (x^{2} y - z) + cos (x^{2} - y z)$

In each case, treat all variables as constants except the one whose partial derivative you are calculating.

$\begin{matrix} \frac{\partial f}{\partial x} & = \frac{\partial}{\partial x} [\frac{x^{2} y - 4 x z + y^{2}}{x - 3 y z}] \\ = \frac{\frac{\partial}{\partial x} (x^{2} y - 4 x z + y^{2}) (x - 3 y z) - (x^{2} y - 4 x z + y^{2}) \frac{\partial}{\partial x} (x - 3 y z)}{{(x - 3 y z)}^{2}} \\ = \frac{(2 x y - 4 z) (x - 3 y z) - (x^{2} y - 4 x z + y^{2}) (1)}{{(x - 3 y z)}^{2}} \\ = \frac{2 x^{2} y - 6 x y^{2} z - 4 x z + 12 y z^{2} - x^{2} y + 4 x z - y^{2}}{{(x - 3 y z)}^{2}} \\ = \frac{x^{2} y - 6 x y^{2} z - 4 x z + 12 y z^{2} + 4 x z - y^{2}}{{(x - 3 y z)}^{2}} \end{matrix}$
$\begin{matrix} \frac{\partial f}{\partial y} & = \frac{\partial}{\partial y} [\frac{x^{2} y - 4 x z + y^{2}}{x - 3 y z}] \\ = \frac{\frac{\partial}{\partial y} (x^{2} y - 4 x z + y^{2}) (x - 3 y z) - (x^{2} y - 4 x z + y^{2}) \frac{\partial}{\partial y} (x - 3 y z)}{{(x - 3 y z)}^{2}} \\ = \frac{(x^{2} + 2 y) (x - 3 y z) - (x^{2} y - 4 x z + y^{2}) (−3 z)}{{(x - 3 y z)}^{2}} \\ = \frac{x^{3} - 3 x^{2} y z + 2 x y - 6 y^{2} z + 3 x^{2} y z - 12 x z^{2} + 3 y^{2} z}{{(x - 3 y z)}^{2}} \\ = \frac{x^{3} + 2 x y - 3 y^{2} z - 12 x z^{2}}{{(x - 3 y z)}^{2}} \end{matrix}$
$\begin{matrix} \frac{\partial f}{\partial z} & = \frac{\partial}{\partial z} [\frac{x^{2} y - 4 x z + y^{2}}{x - 3 y z}] \\ = \frac{\frac{\partial}{\partial z} (x^{2} y - 4 x z + y^{2}) (x - 3 y z) - (x^{2} y - 4 x z + y^{2}) \frac{\partial}{\partial z} (x - 3 y z)}{{(x - 3 y z)}^{2}} \\ = \frac{(−4 x) (x - 3 y z) - (x^{2} y - 4 x z + y^{2}) (−3 y)}{{(x - 3 y z)}^{2}} \\ = \frac{−4 x^{2} + 12 x y z + 3 x^{2} y^{2} - 12 x y z + 3 y^{3}}{{(x - 3 y z)}^{2}} \\ = \frac{−4 x^{2} + 3 x^{2} y^{2} + 3 y^{3}}{{(x - 3 y z)}^{2}} \end{matrix}$
$\begin{matrix} \frac{\partial f}{\partial x} & = \frac{\partial}{\partial x} [sin (x^{2} y - z) + cos (x^{2} - y z)] \\ = (cos (x^{2} y - z)) \frac{\partial}{\partial x} (x^{2} y - z) - (sin (x^{2} - y z)) \frac{\partial}{\partial x} (x^{2} - y z) \\ = 2 x y cos (x^{2} y - z) - 2 x sin (x^{2} - y z) \\ \frac{\partial f}{\partial y} & = \frac{\partial}{\partial y} [sin (x^{2} y - z) + cos (x^{2} - y z)] \\ = (cos (x^{2} y - z)) \frac{\partial}{\partial y} (x^{2} y - z) - (sin (x^{2} - y z)) \frac{\partial}{\partial y} (x^{2} - y z) \\ = x^{2} cos (x^{2} y - z) + z sin (x^{2} - y z) \\ \frac{\partial f}{\partial z} & = \frac{\partial}{\partial z} [sin (x^{2} y - z) + cos (x^{2} - y z)] \\ = (cos (x^{2} y - z)) \frac{\partial}{\partial z} (x^{2} y - z) - (sin (x^{2} - y z)) \frac{\partial}{\partial z} (x^{2} - y z) \\ = − cos (x^{2} y - z) + y sin (x^{2} - y z) \end{matrix}$

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Source: OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2

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