Question

Use the chain rule to find ∂z/∂s and ∂z/∂t for the given equations z = ex 5y, x = s/t, y = t/s?

Answer 1

To use the chain rule, we can find the partial derivatives of x and y with respect to s and t, and then differentiate z with respect to x and y. Finally, we can apply the chain rule to find ∂z/∂s and ∂z/∂t.

Step-by-step explanation:

To find ∂z/∂s and ∂z/∂t using the chain rule, we can start by finding ∂z/∂x, ∂x/∂s, ∂x/∂t, ∂z/∂y, ∂y/∂s, and ∂y/∂t.

Using the given equations:

• x = s/t
• y = t/s
• z = ex^5y

Let's differentiate each equation with respect to the corresponding variable:

• ∂x/∂s = 1/t
• ∂x/∂t = -s/t^2
• ∂y/∂s = -t/s^2
• ∂y/∂t = 1/s
• ∂z/∂x = 5ex^5y
• ∂z/∂y = ex^5

Finally, we can apply the chain rule to find ∂z/∂s and ∂z/∂t:

• ∂z/∂s = (∂z/∂x)(∂x/∂s) + (∂z/∂y)(∂y/∂s) = 5ex^5y * (1/t) + ex^5 * (-t/s^2)
• ∂z/∂t = (∂z/∂x)(∂x/∂t) + (∂z/∂y)(∂y/∂t) = 5ex^5y * (-s/t^2) + ex^5 * (1/s)