Final answer:
The first partial derivatives of the function f(x, t) = t⁴e⁻¹ are found by treating one variable as a constant and differentiating with respect to the other. The result is f_x(x, t) = -t⁴e⁻¹ and f_t(x, t) = 4t³e⁻¹.
Step-by-step explanation:
To find the first partial derivatives of the function f(x, t) = t⁴e⁻¹, we will differentiate the function with respect to each variable while treating the other variable as a constant.
Finding the partial derivative with respect to x:
The partial derivative of f(x, t) with respect to x is found using the chain rule because the function contains an exponential function of x. It is calculated as follows:
f_x(x, t) = ∂f/∂x = t⁴(-e⁻¹) = -t⁴e⁻¹.
Finding the partial derivative with respect to t:
The partial derivative of f(x, t) with respect to t is found by differentiating the polynomial part of the function with respect to t. It is calculated as:
f_t(x, t) = ∂f/∂t = 4t³e⁻¹.
In conclusion, the first partial derivatives of the function are:
- f_x(x, t) = -t⁴e⁻¹
- f_t(x, t) = 4t³e⁻¹