PRACTICE PROBLEMS ON PARTIAL DERIVATIVES WITH SOLUTION

Problem 1 :

If

u(x, y) = x²y+3xy⁴, x = e^t 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+e^2t

find du/dt              Solution

Problem 3 :

If

w(x, y, z) = x²+y²+z², x = e^t, y = e^t sint and z = e^tcost

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 = e^s, y = cos s, s ∈ ℝ

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

Problem 6 :

If

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

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

Problem 7 :

Let

U(x, y) = e^x 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 = se^t, 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).

Solution

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