MAS152 Essential Mathematical Skills & Techniques Examples 4: Partial Differentiation 1. For each of the following functions, calculate ∂f ∂2f ∂2f ∂f and . Show that = . ∂x ∂y ∂x∂y ∂y∂x (a) x3 + 3x2 y + xy 2 + 4y 3 (b) x sin(xy) (c) xy 2 ln(x2 + y 2 ) sin r (d) , where r 2 = x2 + y 2 . r [First show that ∂r/∂x = x/r.] 2. If f (x, y) = 3x2 y + y 3 + 4xy 2 show that x 3. If f (x, y) = cos3 (x2 − y 2 ) find 4. If V = x t3/2 5. If z = ln exp ∂f ∂f +y = 3f. ∂x ∂y ∂f ∂f ∂f ∂f and . Show that y +x = 0. ∂x ∂y ∂x ∂y ∂V x2 ∂2V , show that the ratio of to is a constant. t ∂t ∂x2 p x2 + y 2 , show that ∂2z ∂2z + = 0. ∂x2 ∂y 2 [This equation is called Laplace’s equation in two dimensions. It arises in many physical and engineering applications.] F ′ (r)x ∂f = . 6. Show that if r = x + y and f (x, y) = F (r) then ∂x r ∂2f x2 y2 ′′ ′ Show further that = F (r) + F (r) and hence that ∂x2 r2 r3 2 2 ∂ f ∂ f + = F ′′ (r) + F ′ (r)/r. ∂x2 ∂y 2 2 2 2 1 Answers ∂f ∂x ∂f (b) ∂x ∂f (c) ∂x ∂f (d) ∂x ∂f = 3x2 + 2xy + 12y 2 ∂y ∂f = sin(xy) + xy cos(xy), = x2 cos(xy) ∂y 2x2 y 2 ∂f 2xy 3 2 2 = y 2 ln(x2 + y 2) + 2 , = 2xy ln(x + y ) + x + y 2 ∂y x2 + y 2 x ∂f y = 3 (r cos r − sin r), = 3 (r cos r − sin r) r ∂y r 2 1 ∂2V 3 −5/2 x ∂V 3 −7/2 =− = − exp xt +x t 4. ∂t t 2 4 ∂x2 1. (a) = 3x2 + 6xy + y 2, 2