Question

Find the differential of the function z = e⁽⁻⁴ˣ⁾ cos(5t)

Answer 1

The differential of the function z = e^(-4x) cos(5t) is -4e^(-4x)cos(5t)dx - 5e^(-4x)sin(5t)dt.

Step-by-step explanation:

The differential of the function z = e^(-4x) cos(5t) can be found by taking the partial derivative of z with respect to each variable, x and t.

1. To find the partial derivative with respect to x, we differentiate e^(-4x) with respect to x, which gives us -4e^(-4x). Since cos(5t) does not involve x, we treat it as a constant. So, the partial derivative of z with respect to x is -4e^(-4x)cos(5t).
2. To find the partial derivative with respect to t, we differentiate cos(5t) with respect to t, which gives us -5sin(5t). Since e^(-4x) does not involve t, we treat it as a constant. So, the partial derivative of z with respect to t is -5e^(-4x)sin(5t).

Therefore, the differential of the function z = e^(-4x) cos(5t) is -4e^(-4x)cos(5t)dx - 5e^(-4x)sin(5t)dt.