Michael Corral: Vector Calculus

72 CHAPTER 2. FUNCTIONS OF SEVERAL VARIABLES
We will often simply write ∂f
∂x and∂f ∂y
instead of∂f (x,y) and∂f
∂x ∂y (x,y).
Example 2.11. Find ∂f
∂x and∂f ∂y for the function f(x,y)= sin(xy2 )
x 2 +1 .
Solution: Treating y as a constant and differentiating f(x,y) with respect to x gives
∂f
∂x = (x2 +1)(y 2 cos(xy 2 ))−(2x) sin(xy 2 )
(x 2 +1) 2
and treating x as a constant and differentiating f(x,y) with respect to y gives
∂f
∂y = 2xy cos(xy2 )
x 2 .
+1
Since both ∂f
∂x and∂f are themselves functions of x and y, we can take their partial
∂y
derivatives with respect to x and y. This yields the higher-order partial derivatives:
∂ 2 f ∂ ( ∂f
)
∂x 2= ∂x ∂x
∂ 2 f
∂y∂x = ∂ ( ∂f
)
∂y ∂x
∂ 3 f ∂ ( ∂ 2 f
)
∂x 3= ∂x ∂x 2
∂ 3 f ∂ ( ∂ 2 f
)
∂y∂x 2= ∂y ∂x 2
∂ 3 f
∂y 2 ∂x = ∂ ( ∂ 2 f
)
∂y ∂y∂x
∂ 3 f
∂x∂y∂x = ∂ ( ∂ 2 f
)
∂x ∂y∂x
∂ 2 f ∂ ( ∂f
)
∂y 2= ∂y ∂y
∂ 2 f
∂x∂y = ∂ ( ∂f
)
∂x ∂y
∂ 3 f ∂ ( ∂ 2 f
)
∂y 3= ∂y ∂y 2
∂ 3 f ∂ ( ∂ 2 f
)
∂x∂y 2= ∂x ∂y 2
∂ 3 f
∂x 2 ∂y = ∂ ∂x
∂ 3 f
∂y∂x∂y = ∂ ∂y
( ∂ 2 f
)
∂x∂y
( ∂ 2 f
)
∂x∂y
.
Example 2.12. Find the partial derivatives ∂f
∂x ,∂f f f ∂ 2 f
∂y ,∂2 ∂x 2,∂2 ∂y 2, ∂y∂x and ∂2 f
∂x∂y
function f(x,y)=e x2y + xy 3 .
for the