Lesson 19: Partial Derivatives (worksheet Solutions)

Solutions to Worksheet for Lesson 19 (Section 15.3 and 15.5) Partial Derivatives Math 20 November 2, 2007 Find all first partial derivatives of the functions. 1. f (x, y) = 3x + 2xy 2 − 2y 4 Solution. We have ∂f = 3 + 2y 2 ∂x

∂f = 4xy − 8y 3 ∂y

∂w = cos α cos β ∂α

∂w = − sin α sin β ∂β

2. w = sin α cos β Solution. We have

3. f (u, v) = arctan(u/v) (Remember that

d 1 arctan(x) = .) dx 1 + x2

Solution. For this it’s important to remember the chain rule! 1 ∂ u 1 1 ∂f = = 2 2 ∂u 1 + (u/v) ∂u v 1 + (u/v) v ∂f 1 ∂ u 1 −u = = ∂u 1 + (u/v)2 ∂v v 1 + (u/v)2 v 2 Another way to write this is ∂f −u = 2 ∂v u + v2

∂f v = 2 ∂u u + v2

1

4. u =

q

x21 + x22 + · · · + x2n

Solution. We have a partial derivative for each index i, but luckily they’re symmetric. So each derivative is represented by: ∂u ∂ 1 = p 2 (x21 + x22 + · · · + x2n ) 2 2 ∂xi ∂x 2 x1 + x2 + · · · + xn i xi =p 2 2 x1 + x2 + · · · + x2n

5. Find all the second derivatives of the functions in Problems 1 and 2. Solution. For Problem 1, we have ∂2f = 4y ∂x∂y ∂2f = −24y 2 ∂y 2

∂2f =0 ∂x2 ∂2f = 4y ∂y∂x For Problem 2, we have

∂ cos α cos β = − sin α cos β ∂α = − cos α sin β ∂ sin α sin β = − sin α cos β =− ∂β

wαα = wβα = wαβ wββ

00 00 6. Verify that f12 = f21 in Problem 3.

Solution. We have (u2 + v 2 )(1) − v(2v) u2 − v 2 = 2 2 2 2 (u + v ) u + v 2 )2 (u2 + v 2 )(−1) − (−u)(2v) u2 − v 2 00 f21 (v, u) = = (u2 + v 2 )2 u2 + v 2 )2

00 f12 (u, v) =

2