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Steps in Finding Partial Derivatives

How to compute partial derivatives of a function of two or more variables:

  1. Identify which variable you are differentiating with respect to (usually $x$, $y$, or $z$).
  2. Treat all other variables as constants. Only the chosen variable is allowed to change.
  3. Differentiate the function using ordinary differentiation rules (power rule, product rule, chain rule, etc.).
  4. Write the result using partial derivative notation such as $\frac{\partial f}{\partial x}$ or $f_x$.
  5. If needed, repeat the process to find higher-order partial derivatives like $f_{xx}$, $f_{yy}$, or mixed partials such as $f_{xy}$ and $f_{yx}$.

Notation for partial derivatives:

$$f_x = \frac{\partial f}{\partial x}$$
$$f_y = \frac{\partial f}{\partial y}$$
$$f_{xx} = \frac{\partial^2 f}{\partial x^2}$$
$$f_{xy} = \frac{\partial^2 f}{\partial x \partial y}$$

Partial derivatives appear in problems involving surfaces, optimization of functions of two variables, and models where a quantity depends on more than one input.

Basic Partial Derivatives — CE Board

Find $f_x$ and $f_y$ if $f(x,\,y) = x^3 + 2x^2y - 3y^2$.

For $f_x$: Treat $y$ as a constant, differentiate with respect to $x$.

$f_x = 3x^2 + 4xy$

For $f_y$: Treat $x$ as a constant, differentiate with respect to $y$.

$f_y = 2x^2 - 6y$

Mixed Partial Derivative — CE Board

Find $f_{xy}$ if $f(x,\,y) = x^3y^2 + 2xy$.

Step 1: Find $f_x$ (differentiate with respect to $x$, $y$ constant):

$f_x = 3x^2y^2 + 2y$

Step 2: Differentiate $f_x$ with respect to $y$:

$f_{xy} = \dfrac{\partial}{\partial y}(3x^2y^2 + 2y) = 6x^2y + 2$

Partial Derivatives of Exponential — CE Board

Find $\dfrac{\partial z}{\partial x}$ and $\dfrac{\partial z}{\partial y}$ if $z = e^{xy}$.

$\dfrac{\partial z}{\partial x} = y\,e^{xy}$
$\dfrac{\partial z}{\partial y} = x\,e^{xy}$

In both cases the chain rule is applied; the other variable acts as a constant multiplier in the exponent.

All Second Partial Derivatives — CE Board

If $f(x,\,y) = x^2 + 3xy + y^2$, find $f_{xx}$, $f_{yy}$, $f_{xy}$, and $f_{yx}$.

First partial derivatives:

$f_x = 2x + 3y,\quad f_y = 3x + 2y$

Second partial derivatives:

$f_{xx} = 2,\quad f_{yy} = 2,\quad f_{xy} = 3,\quad f_{yx} = 3$

Note: $f_{xy} = f_{yx}$ (Clairaut's theorem — mixed partials are equal for smooth functions).

Applied Partial Derivative — Cylinder (CE Board)

The volume of a cylinder is $V = \pi r^2 h$. Find $\dfrac{\partial V}{\partial r}$ and $\dfrac{\partial V}{\partial h}$, and interpret each result.

$\dfrac{\partial V}{\partial r} = 2\pi r h$

This is the rate of change of volume with respect to radius (height held constant).

$\dfrac{\partial V}{\partial h} = \pi r^2$

This is the rate of change of volume with respect to height (radius held constant) — equal to the base area of the cylinder.

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