In Exercises 41–58, find dy/dt.


y = √(3t + (√2 + √(1 − t)))

Verified step by step guidance

1. Identify the function y in terms of t: y = √(3t + (√2 + √(1 − t))).
2. Apply the chain rule to differentiate y with respect to t. The chain rule states that if a function y = f(g(t)), then dy/dt = f'(g(t)) * g'(t).
3. First, differentiate the outer function, which is the square root: If u = 3t + (√2 + √(1 − t)), then dy/du = 1/(2√u).
4. Next, differentiate the inner function u = 3t + (√2 + √(1 − t)) with respect to t. This involves differentiating each term separately: du/dt = 3 + d/dt(√2) + d/dt(√(1 − t)).
5. For the term √(1 − t), use the chain rule again: Let v = 1 − t, then d/dt(√v) = (1/2√v) * (-1). Combine all these derivatives to find du/dt, and then multiply by dy/du to find dy/dt.

Key Concepts

Chain Rule

The chain rule is a fundamental technique in calculus used to differentiate composite functions. It states that if a function y = f(g(t)) is composed of two functions, the derivative dy/dt is found by multiplying the derivative of the outer function f with respect to the inner function g by the derivative of the inner function g with respect to t. This rule is essential for differentiating the given function y = √(3t + (√2 + √(1 − t))).

Derivative of Square Root Function

The derivative of a square root function, such as √u, is given by (1/2√u) * du/dt, where u is a function of t. This formula is crucial for differentiating the outermost square root in the given function y = √(3t + (√2 + √(1 − t))). Understanding how to apply this derivative is key to solving the problem.

Implicit Differentiation

Implicit differentiation is a technique used when a function is not explicitly solved for one variable in terms of another. In this problem, it helps differentiate nested functions like √(1 − t) within the composite function. By treating y as a function of t and applying the chain rule, implicit differentiation allows us to find dy/dt even when y is not isolated.