Final answer:
To calculate the derivative of f(t) = √(5 - t), we use the chain rule by taking the derivative of the outer function, which is the square root function, and multiplying it by the derivative of the inner function, which is (5 - t). The final answer is f'(t) = -1/(2√(5 - t)).
Step-by-step explanation:
The student is asking to find the derivative of the function f(t) = √(5 - t) using the chain rule. To find the derivative, identify the outer function (the square root) and the inner function (5 - t). Then, take the derivative of the outer function with respect to the inner function, followed by the derivative of the inner function with respect to t. The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function.
Let's denote the inner function as g(t) = 5 - t and the outer function as h(g) which is the square root of g, i.e., h(g) = √(g). The derivative of the outer function h with respect to g, h'(g), is 1/(2√(g)) and the derivative of the inner function g with respect to t is -1.
Applying the chain rule, we get:
f'(t) = h'(g(t)) × g'(t) = ¹/₂ × 1/√(5 - t) × (-1) = -1/(2√(5 - t))