Question

Use the chain rule to find dz/dt. Given z = (x - y)/(x⁴y), where x = eᵗ and y = e⁽⁻ᵗ⁾.

Answer 1

To find dz/dt for the function z = (x - y)/(x⁴y) with x = e^t and y = e^(-t), calculate the partial derivatives dz/dx and dz/dy, then find dx/dt and dy/dt, and apply the chain rule by multiplying and adding the respective derivatives.

Step-by-step explanation:

To solve for dz/dt using the chain rule, we need to find the partial derivatives of z with respect to x and y, respectively, and then multiply each by the derivative of x and y with respect to t. Given that z = (x - y)/(x⁴y) and x = et, y = e-(t), we follow these steps:

1. Calculate the partial derivatives dz/dx and dz/dy:

• dz/dx = (y - 4x3(x - y))/(x⁴y2)
• dz/dy = -(1)/(x⁰) + (x - y)/(x⁴y2)

Find the derivatives dx/dt and dy/dt:

• dx/dt = et
• dy/dt = -e-(t)

Apply the chain rule to find dz/dt:

• dz/dt = (dz/dx)(dx/dt) + (dz/dy)(dy/dt)
• Substitute the derivatives obtained in the steps above and simplify to get the final expression for dz/dt.