Question

Suppose f(x,t) = x⁴ * sin(4t). At any point (x, t), the differential is?

Answer 1

The differential of the function f(x,t) = x⁴ ∗ sin(4t) is given by the sum of its partial derivatives, resulting in df = (4x³ ∗ sin(4t))dx + (4x⁴ ∗ cos(4t))dt.

Step-by-step explanation:

To find the differential of the function f(x,t) = x⁴ ∗ sin(4t), we need to compute the partial derivatives with respect to each variable (x and t) individually. The partial derivative of f with respect to x while holding t constant is 4x³ ∗ sin(4t). The partial derivative of f with respect to t while holding x constant is x⁴ ∗ cos(4t) ∗ 4. The total differential df can then be expressed as df = (∂f/∂x)dx + (∂f/∂t)dt = (4x³ ∗ sin(4t))dx + (4x⁴ ∗ cos(4t))dt.