# Differentials of multivariable functions

### Differentials of multivariable functions

#### Lessons

Notes:

Differentials of 2 Variable Functions
Suppose there is a 2-variable function $z=f(x,y)$. Then we say that the differential $dz$ is:

$dz = f_xdx + f_ydy$

Differentials of 3 Variable Functions
Of course, differentials can be extended to 3-variable functions as well. Suppose there is a 3-variable function $w=g(x,y, z)$. Then we say that the differential $dw$ is:

$dw = g_x dx + g_y dy + g_z dz$

• Introduction
Differentials of Multivariable Functions Overview:
a)
A Review of Differentials
• $dy, dx$ are differentials
• $dy=f'(x)dx$

b)
Differentials of 2 Variable Functions
• Differential $dz$
• $dz=f_xdx+f_ydy$
• An example

c)
Differentials of 3 Variable Functions
• Differential $dw$
• $dw=g_xdx+g_ydy+g_zdz$
• An example

• 1.
Finding Differentials of 2 Variable Functions
Compute the differential for the following function:

$f(x,y) = e^{x^3+y^3} \sin \frac{x}{2}$

• 2.
Compute the differential for the following function:

$z = ln (\frac{x^2 y^3 }{2} )$

• 3.
Finding Differentials of 3 Variable Functions
Compute the differentials for the following function:

$g(x,y,z) = \tan [ \ln (xy^2 z^3) ]$

• 4.
Compute the differentials for the following function:

$g = \frac{e^{xy}}{xy^2z^2}$