Final answer:
To find the differential of the function z = e^{-4x} cos(6\pi t), differentiate partially with respect to both x and t using the product rule and chain rule, resulting in dz = -4e^{-4x} cos(6\pi t) dx - 6\pi e^{-4x} sin(6\pi t) dt.
Step-by-step explanation:
To find the differential of the function z = e^{-4x} \cos(6\pi t), we need to use the product rule and chain rule to differentiate with respect to both x and t.
The differential dz is composed of two parts: one for x and one for t. Calculating the partial derivative with respect to x, we get:
d( e^{-4x} \cos(6\pi t) )/dx = \cos(6\pi t) \cdot d( e^{-4x})/dx + e^{-4x} \cdot d(\cos(6\pi t))/dx
Since \cos(6\pi t) is not a function of x, its derivative with respect to x is 0, and we only need to differentiate e^{-4x}, which gives us -4e^{-4x} \cos(6\pi t). Then, we have:
\partial z/\partial x = -4e^{-4x} \cos(6\pi t)
Similarly, we find the partial derivative with respect to t:
d( e^{-4x} \cos(6\pi t) )/dt = e^{-4x} \cdot d(\cos(6\pi t))/dt + \cos(6\pi t) \cdot d( e^{-4x})/dt
Here, e^{-4x} is a constant with respect to t, so its derivative is 0, and we need to differentiate \cos(6\pi t) with respect to t, obtaining -6\pi e^{-4x} \sin(6\pi t). Thus:
\partial z/\partial t = -6\pi e^{-4x} \sin(6\pi t)
In conclusion, the differential dz for the function z is:
dz = -4e^{-4x} \cos(6\pi t) dx - 6\pi e^{-4x} \sin(6\pi t) dt