Partial Derivatives formulas with some examples

 How to find partial derivatives?

--> 1 ) If z = f(x, y) is a function of two real variables x and y, then partial derivative of z w.r.to x is denoted by

and is the ordinary derivative of z w.r.to x by keeping y constant/fixed.

  2)Similarly partial derivative of z w.r.to y is denoted by

and is the ordinary derivative of z w.r.to y by keeping x constant/fixed.

Note :

                              (3)If f x, y, z is a function of three variables x, y and z then partial

                              derivative of f w.r.to any single variable is obtained by treating

                               remaining all variables constant.

                           (4) All the usual rules for differentiating sums, differences, products,

                               quotients and functions of a function obeys in partial derivatives.

                                 

Examples:

1. Find the values of ∂f/∂x and ∂f/∂y

at the point (4, −5) if: f(x, y) = x^2 + 3xy + y − 1

Solution: To find ∂f/∂x ,treat y as a constant and differentiate with respect to x

∴∂f/∂x = 2x + 3y

To find ∂f/∂y ,treat x as a constant and differentiate with respect to y

∴ ∂f/∂y = 3x + 1

⟹(∂f/∂x)(4,−5) = 2* 4 + 3 −5 = −7

     (∂f/∂y)(4,−5) = 3 *4 + 1 = 13

2. Find the values of ∂f/∂x and ∂f/∂y where f x, y = y sin (xy) .

Solution: f x, y = y sin (xy )... ... ... (1)

Differentiate (1) w.r.to x by treating y as a constant

∴ ∂f/∂x = y cos xy × y = y^2 cos(xy)

Differentiate (1) w.r.to y by treating x as a constant

∴ ∂f/∂y = sin xy + ycos(xy) × x 

            = sin xy + xy cos (xy)