Unizor - Partial Derivatives - Examples

Notes to a video lecture on http://www.unizor.com

Partial Derivatives - Examples

Example 1

Let z=√x·y
Then
∂z/∂x = assuming y constant = y/(2√x·y)
∂z/∂y = assuming x constant = x/(2√x·y)

Example 2

Let z=e^x·y
Then
∂z/∂x = assuming y constant = y·e^x·y
∂z/∂y = assuming x constant = x·e^x·y

Example 3

Let z=1/(x²+y²)
Then
∂z/∂x = assuming y constant = −2x/(x²+y²)²
∂z/∂y = assuming x constant = −2y/(x²+y²)²

Example 4

Let z=sin(x)/y²
Then
∂z/∂x = assuming y constant = cos(x)/y²
∂z/∂y = assuming x constant = −2sin(x)/y³

Example 5

Let z=arctan(x√y)
Then
∂z/∂x = assuming y constant = √y/(1+x²·y)
∂z/∂y = assuming x constant = x/[2√y(1+x²·y)]

Example 6

Let z=y^x
Then
∂z/∂x = assuming y constant = y^x·ln(y)
∂z/∂y = assuming x constant = x·y^x−1