Use the Chain Rule to find ∂z/∂s and ∂z/∂t.
z = (x − y)5, x = s2t, y = st2
Solution:
z = (x − y)5, x = s2t, y = st2 (Given)
To find:
∂z/∂s and ∂z/∂t.
As per the chain rule of partial differentiation, we have
∂z/∂s = ∂z/∂x . ∂x/∂s + ∂z/∂y . ∂y/∂s --------> (1)
∂z/∂t = ∂z/∂x . ∂x/∂t + ∂z/∂y . ∂y/∂t---------> (2)
Applying partial derivatives,
With y as constant and differentiating with respect to x
∂z/∂x = 5 (x - y)4 (1) = 5 (x - y)4
With x as constant and differentiating with respect to y
∂z/∂y = 5 (x - y)4 (-1) = - 5 (x - y)4
With t as constant and differentiating with respect to s
∂x/∂s = 2st and ∂y/∂s = t2
Now substitute the values in (1)
∂z/∂s = [5 (x - y)4] (2st) + [- 5 (x - y)4] t2
∂z/∂s = 5 (x - y)4 (2st - t2)
With s as constant and differentiating with respect to t
∂x/∂t = s2 and ∂y/∂t = 2st
Now substitute the values in (2)
∂z/∂t = [5 (x - y)4] (s2) + [- 5 (x - y)4] (2st)
∂z/∂t = 5 (x - y)4 (s2 - 2st)
Therefore, ∂z/∂s = 5 (x - y)4 (2st - t2) and ∂z/∂t = 5 (x - y)4 (s2 - 2st).
Use the Chain Rule to find ∂z/∂s and ∂z/∂t.
z = (x − y)5, x = s2t, y = st2
Summary:
Using the Chain Rule, ∂z/∂s = 5 (x - y)4 (2st - t2) and ∂z/∂t = 5 (x - y)4 (s2 - 2st) for z = (x − y)5, x = s2t, y = st2.
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