Partial Derivatives

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Function f(x,y) of two variables, there are two partial derivatives ∂f/∂x and ∂f/∂y.

∂f/∂x (a,b) = lim(h→0) f( a+h , b ) - f( a , b )
h
∂f/∂y (a,b) = lim(h→0) f( a , b+k ) - f( a , b )
k
  • For ∂f/∂x , f is differentiated as a function with respect to x and y is treated as a constant.
  • For ∂f/∂x , f is differentiated as a function with respect to y and x is treated as a constant.

Example 1 :

Solve for ∂f/∂x and ∂f/∂y if f(x,y) = .
Solution :
∂f/∂x = -2x
∂f/∂y = 1

Example 2 :

Solve for ∂g/∂x (3,2) and ∂g/∂y (3,2) if g(x,y) = xye^.
Solution :
∂g/∂x = (xy)*(2xe^) + ye^
= 2ye^ + ye^ =ye^(2+1)
∂g/∂x (3,2) = 2e^(2+1)
= 2(19)
= 38


∂g/∂y = xe^
∂g/∂y (3,2) = 3



Example 3 :

Solve for ∂f/∂x and ∂f/∂y if f(x,y)= ln(xy) +sin(x) = ln(x) + ln(y)+ sin(x).
Solution :
∂f/∂x =
∂f/∂y =

Example 4 :

Find the rectangular box of least surface area that has volume 1000.
Solution :
Volume = 1000 = xyz
Surface Area = 2xz + 2yz + 2xy
z = 1000/xy
S = 2x(1000/xy) + 2y(1000/xy) + 2xy
= 2000/y + 2000/x + 2xy *need to minimize this*
Solve for ∂S/∂x = 0 and ∂S/∂y = 0
∂S/∂x = -2000/ + 2y = 0
∂S/∂y = -2000/ + 2x = 0
y = 1000/ and x = 1000/
y() = 0
Therefore, y = 0 or -1000 = 0
So, y = 10 and x = 10.
Using the second derivative test, S does have a local minimum at (10, 10).