Final answer:
To find the differential of the function z = e⁻⁴ˣ cos 6πt, take the derivative of z with respect to x and t separately. The differential with respect to x is -4e⁻⁴ˣ * cos 6πt * dx. The differential with respect to t is e⁻⁴ˣ * (-6πsin 6πt) * dt.
Step-by-step explanation:
To find the differential of the function z = e^(-4x) cos(6πt), we need to take its derivative with respect to x and t separately. Let's start with the derivative with respect to x. The derivative of e^(-4x) is -4e^(-4x), and the derivative of cos(6πt) is 0. Therefore, the differential of z with respect to x is -4e^(-4x) * cos(6πt) * dx.
Now let's find the differential with respect to t. The derivative of e^(-4x) is 0, and the derivative of cos(6πt) is -6πsin(6πt). So, the differential of z with respect to t is e^(-4x) * (-6πsin(6πt)) * dt. Combining the two differentials, we have dz = -4e^(-4x) * cos(6πt) * dx + e^(-4x) * (-6πsin(6πt)) * dt."