True or False? In Exercises 137-139, determine whether the...

True or False? In Exercises 137-139, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
If $y=(1-x)^{1 / 2},$ then $y^{\prime}=\frac{1}{2}(1-x)^{-1 / 2}$

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Key Concepts

Chain Rule

The chain rule is a fundamental technique used to differentiate composite functions. It states that when a function is composed of two functions, the derivative of the composite function is the derivative of the outer function evaluated at the inner function multiplied by the derivative of the inner function. In this context, it is crucial for correctly differentiating a function like (1-x)^(1/2).

Power Rule

The power rule is a basic differentiation technique that applies to functions of the form f(x)=x^n, yielding a derivative of n*x^(n-1). When a function is expressed in a power form, even with a composite argument, the power rule is used in conjunction with the chain rule to differentiate the overall function. Here, it is applied to the exponent 1/2 in (1-x)^(1/2).

Composite Function Differentiation

Differentiating composite functions requires identifying the inner and outer functions and applying the chain rule along with other differentiation rules like the power rule. This concept is essential for accurately calculating derivatives when the function consists of multiple layers, as seen in the expression (1-x)^(1/2), where the derivative of the inner function (1-x) must be considered.

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