Final answer:
To find dw/dt, substitute expressions for x, y, and z into the function w = xyz, and then apply the Chain Rule by taking the sum of the product of partial derivatives of w with respect to x, y, and z, and multiply by the derivatives of those variables with respect to t.
Step-by-step explanation:
To calculate dw/dt for the given function w = xyz, where x = t², y = 7t, and z = eᵗ, we can use the Chain Rule for differentiation. First, we express w in terms of t by substituting the expressions for x, y, and z. Then, we differentiate w with respect to t, treating x, y, and z as composite functions of t.
The derivative dw/dt will be the sum of partial derivatives of w with respect to each variable (x, y, and z) multiplied by the derivatives of those variables with respect to t respectively. This can be presented as:
dw/dt = (dw/dx)(dx/dt) + (dw/dy)(dy/dt) + (dw/dz)(dz/dt)
Substitute the known expressions to find the derivative:
dw/dt = (yz)(2t) + (xz)(7) + (xy)(eᵗ)
Now plug in the expressions for x, y, and z and simplify to obtain the final result of dw/dt.