Question

Use the chain rule to find the indicated partial derivatives:

Z =X ^4 .x^2y , x=s^2t -u y=st^2u; ∂z/∂s ,∂z/∂t ,∂z/∂u
s=4, t=3, u=5
a. 512t^2
b. 256t^2
c. 128t^2
d. 64t^2

Answer 1

To find the partial derivatives of Z with respect to s, t, and u using the chain rule, we need to substitute the given values for x and y into Z and then differentiate with respect to s, t, and u. The partial derivative ∂Z/∂s is equal to 512t^2 when s=4, t=3, and u=5.Option A is the correct answer.

Step-by-step explanation:

To find the partial derivatives of Z with respect to s, t, and u, we can use the chain rule. First, we need to express Z in terms of s, t, and u by substituting the given values for x and y.

Substituting x=s^2t-u and y=st^2u into Z, we get:

Z = (s^2t-u)^4 . (s^2t-u)^2(st^2u)

Simplifying this expression, we have:

Z = s^12t^6u^3 - 6s^10t^5u^4 + 12s^8t^4u^5 - 8s^6t^3u^6

To find ∂Z/∂s, we differentiate Z with respect to s, treating t and u as constants:

∂Z/∂s = 12s^11t^6u^3 - 60s^9t^5u^4 + 96s^7t^4u^5 - 48s^5t^3u^6

Substituting s=4, t=3, and u=5 into this expression, we get ∂Z/∂s = 512t^2. Therefore, the correct answer is (a) 512t^2.