Partial Derivative

Partial Derivative:

be a function of two variables then we define Let
The partial derivative as:
with respect to x is The partial derivative of

with Similarly, we define the partial derivative of
respect to y

if these limits exists.
Algebraically, we can think of the partial derivative of a function with respect to x as the derivative of the function with y held constant. Geometrically, the derivative with respect to x at a point P represents the slope of the curve that passes through P whose projection onto the xy plane is a horizontal line.



Example:
find Let
Solution:




for the partial derivatives and We also use the notation
with respect to x and y respectively.
Exercise:
for the function from the example above. Find
Finding Partial Derivative the Easy Way:

Since a partial derivative with respect to x is a derivative with the rest of the variables held constant, we can find the partial derivative by taking the regular derivative considering the rest of the variables as constants.


Example:
Find: Let
(2) (1)
Solution:
(1)
(2)

Example:
find: Let
(2) (1)
Solution:
(1)
(2)

Example:
at the point (4,5) if (1) Find
at the point (4,5) if (2) Find

Solution:
(1) We regard y as a constant and differentiate with respect to x :

at the point (4,5) is the value of

(2) We regard x as a constant and differentiate with respect to y :


at the point (4,5) is the value of

Exercise:
(a) Find both partial derivatives for : (1)
(2)
if (b) Find
if and (c) Find