# PRACTICE PROBLEMS ON PARTIAL DERIVATIVES WITH SOLUTION

Problem 1 :

If

u(x, y) = x2y+3xy4, x = et and y = sin t

find du/dt and evaluate it at t = 0.

Solution

Problem 2 :

If

u(x, y, z) = xy2z3, x = sint, y = cost, z = 1+e2t

find du/dt              Solution

Problem 3 :

If

w(x, y, z) = x2+y2+z2, x = et, y = et sint and z = etcost

find dw/dt.                          Solution

Problem 4 :

Let

U(x, y, z) = xyz, x = e-t, y = e-tcost, z = sint, t ℝ.

Find dU/dt.              Solution

Problem 5 :

If

w(x, y) = 6x3-3xy+2y2, x = es, y = cos s, s ∈

find dw/ds, and evaluate at s = 0.          Solution

Problem 6 :

If

z(x,y) = x tan-1(xy), x = t2, y = set, s,t ∈ ℝ.

Find ∂z/∂s and ∂z/∂t at s = t = 1        Solution

Problem 7 :

Let

U(x, y) = ex sin y

where x = st2, y = s2t s,t ∈ ℝ. Find ∂U/∂s, ∂U/∂t and evaluate them at s = t = 1.        Solution

Problem 8 :

Let

z(x, y) = x3-3x2y3

where x = set, y = se-t, s, t  ∈ ℝ. Find ∂z/∂s, ∂z/∂t.

Solution

Problem 9 :

W(x, y, z) = xy + yz + zx, x = u − v, y = uv, z = u + v, u,v  ∈ ℝ. Find ∂W/∂u, ∂W/∂v and evaluate them at (1/2, 1).

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