True or False? determine whether the statement is true or...

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

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Chain Rule

The chain rule is a fundamental tool in differentiation used when a function is composed of two or more functions. It states that 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. This concept is crucial for correctly differentiating functions like (1-x)^(1/2) because it involves an inner function (1-x) and an outer function (u^(1/2)).

Power Rule

The power rule is a basic differentiation rule that states that the derivative of x^n is n*x^(n-1) for any real exponent n. When applying the power rule to a function raised to a power, it is necessary to combine it with the chain rule if the expression inside the power is not simply x, ensuring that any additional factors arising from the inner function’s derivative are accounted for.

Composite Functions

Composite functions are functions formed by applying one function to the result of another, such as f(g(x)). Differentiating composite functions requires careful application of the chain rule, as failing to do so may lead to an incorrect derivative. In this context, recognizing the structure of the function and applying the appropriate differentiation techniques is essential for obtaining the correct result.

Please give Ace some feedback