Question

Use the Chain Rule to find dz/dt.

z = x² + y² + xy, x = sin t, y = 5eᵗ

dz/dt=

Answer 1

To find dz/dt, we first need to find the derivatives of x and y with respect to t using the chain rule. Then, substitute these values into the expression for dz/dt and simplify.

Step-by-step explanation:

To find dz/dt, we first need to find the derivatives of x and y with respect to t. Given that x = sin(t) and y = 5e^t, we can find dx/dt and dy/dt using the chain rule. The derivative of x with respect to t is dx/dt = cos(t), and the derivative of y with respect to t is dy/dt = 5e^t.

Next, we substitute these values into the expression for dz/dt. Using the chain rule, we have dz/dt = (dz/dx)(dx/dt) + (dz/dy)(dy/dt).

Since z = x^2 + y^2 + xy, we can find the partial derivatives dz/dx and dz/dy. dz/dx = 2x + y and dz/dy = 2y + x. Substituting these values, we have dz/dt = (2x + y)(dx/dt) + (2y + x)(dy/dt).