Find the derivatives of the functions in Exercises 19–40.

p = √(3 − t)

검증된 단계별 안내

Identify the function you need to differentiate: \( p = \sqrt{3 - t} \). This can be rewritten using exponent notation as \( p = (3 - t)^{1/2} \).

Apply the chain rule for differentiation. The chain rule states that if you have a composite function \( f(g(t)) \), then the derivative \( f'(g(t)) \cdot g'(t) \) is used.

Differentiate the outer function \( f(u) = u^{1/2} \) with respect to \( u \). The derivative is \( \frac{1}{2}u^{-1/2} \).

Differentiate the inner function \( g(t) = 3 - t \) with respect to \( t \). The derivative is \( -1 \).

Combine the derivatives using the chain rule: \( \frac{d}{dt} p = \frac{1}{2}(3 - t)^{-1/2} \cdot (-1) \). Simplify the expression to find the derivative of \( p \).

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주요 개념

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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 practical terms, the derivative provides the slope of the tangent line to the function's graph at any given point.

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 u and t, then the derivative of y with respect to t can be found by multiplying the derivative of y with respect to u by the derivative of u with respect to t. This rule is essential for functions that are nested within one another.

Square Root Function

The square root function, denoted as √x, is a function that returns the non-negative value whose square is x. When differentiating functions involving square roots, it is important to apply the chain rule correctly, as the square root can be expressed as a power of one-half. Understanding the properties of square roots is crucial for accurately finding derivatives of such functions.

Find the derivatives of the functions in Exercises 19–40.p = √(3 − t)