Final answer:
To find the derivative of a function f with respect to t when f is a function of both x and t, we use the chain rule. The derivative is given by ∂f / ∂x dx / dt + ∂f / ∂t. Here, ∂ represents the partial derivative and dx / dt represents the derivative of x(t) with respect to t.
Step-by-step explanation:
The question asks for the derivative of a function f with respect to t when f is a function of both x and t. To find this derivative, we use the chain rule. Let's denote f as f(x(t), t). Using the chain rule, the derivative of f with respect to t is given by:
df(x(t), t)
dt = ∂f / ∂x dx / dt + ∂f / ∂t
Here, ∂ represents the partial derivative and dx / dt represents the derivative of x(t) with respect to t. The terms ∂f / ∂x and ∂f / ∂t are the partial derivatives of f with respect to x and t respectively.