27–76. Calculate the derivative of the following functions.
y = √f(x), where f is differentiable and nonnegative at x.

Verified step by step guidance

1. Step 1: Recognize that the function y = \sqrt{f(x)} can be rewritten using exponent notation as y = [f(x)]^{1/2}. This will make it easier to apply the chain rule.
2. Step 2: Apply the chain rule for differentiation. The chain rule states that if you have a composite function y = g(h(x)), then the derivative y' is g'(h(x)) * h'(x).
3. Step 3: Differentiate the outer function g(u) = u^{1/2} with respect to u. The derivative is g'(u) = \frac{1}{2}u^{-1/2}.
4. Step 4: Substitute back the inner function f(x) into the derivative of the outer function. This gives us \frac{1}{2}[f(x)]^{-1/2}.
5. Step 5: Multiply the result from Step 4 by the derivative of the inner function f(x), which is f'(x). Therefore, the derivative of y = \sqrt{f(x)} is \frac{1}{2}[f(x)]^{-1/2} \cdot f'(x).

Key Concepts

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. In calculus, the derivative is often denoted as f'(x) or dy/dx, and it provides critical information about the function's behavior, such as its slope and points of tangency.

Chain Rule

The chain rule is a fundamental technique in calculus used to differentiate composite functions. It states that if a function y is composed of two functions, say y = g(f(x)), then the derivative of y with respect to x can be found by multiplying the derivative of g with respect to f by the derivative of f with respect to x. This rule is essential when dealing with functions that are nested within one another, such as the square root function in the given problem.

Square Root Function

The square root function, denoted as √f(x), is a specific type of function that returns the nonnegative square root of its input. In calculus, it is important to recognize that the square root function is defined only for nonnegative values, which affects its domain. When differentiating this function, one must apply the chain rule and consider the properties of square roots, particularly how they behave as the input approaches zero or becomes negative.