Final answer:
To find ∂z/∂s and ∂z/∂t using the chain rule is ∂z/∂s = 5x⁴y⁵ * cos t + 5x⁵y⁴ * sin t and ∂z/∂t = -5x⁴y⁵ * s sin t + 5x⁵y⁴ * s cos t.
Step-by-step explanation:
To find ∂z/∂s and ∂z/∂t using the chain rule, we need to differentiate z with respect to s and t separately.
Let's start with ∂z/∂s. Using the chain rule, we have:
∂z/∂s = ∂z/∂x * ∂x/∂s + ∂z/∂y * ∂y/∂s
Since z = x⁵y⁵, we can find ∂z/∂x and ∂z/∂y using the power rule:
∂z/∂x = 5x⁴y⁵
∂z/∂y = 5x⁵y⁴
Now, let's find ∂x/∂s and ∂y/∂s. Given x = s cos t and y = s sin t, we can differentiate them with respect to s:
∂x/∂s = cos t
∂y/∂s = sin t
Substituting these values into the chain rule equation, we get:
∂z/∂s = 5x⁴y⁵ * cos t + 5x⁵y⁴ * sin t
Similarly, to find ∂z/∂t, we differentiate z with respect to t:
∂z/∂t = ∂z/∂x * ∂x/∂t + ∂z/∂y * ∂y/∂t
Following the same steps as above, we find:
∂z/∂t = -5x⁴y⁵ * s sin t + 5x⁵y⁴ * s cos t
In summary, using the chain rule, we found ∂z/∂s = 5x⁴y⁵ * cos t + 5x⁵y⁴ * sin t and ∂z/∂t = -5x⁴y⁵ * s sin t + 5x⁵y⁴ * s cos t.