27–76. Calculate the derivative of the following functions.y=10x+1y=\sqrt{10x+1}

Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 3 - Derivatives

문제 3.7.29

3장, 문제 3.7.29

27–76. Calculate the derivative of the following functions.
$y = \sqrt{10 x + 1}$

검증된 단계별 안내

Step 1: Recognize that the function given is $ y = \sqrt{10x + 1} $. This can be rewritten using exponent notation as $ y = (10x + 1)^{1/2} $.

Step 2: To find the derivative, apply the chain rule. The chain rule states that if you have a composite function $ y = f(g(x)) $, then the derivative $ y' = f'(g(x)) \cdot g'(x) $.

Step 3: Identify the outer function $ f(u) = u^{1/2} $ and the inner function $ g(x) = 10x + 1 $.

Step 4: Differentiate the outer function with respect to $ u $: $ f'(u) = \frac{1}{2}u^{-1/2} $.

Step 5: Differentiate the inner function with respect to $ x $: $ g'(x) = 10 $.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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 curve of the function 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 x (i.e., y = f(u) and u = g(x)), then the derivative of y with respect to x can be found by multiplying the derivative of f with respect to u by the derivative of g with respect to x. This rule is essential for handling functions that are nested within each other.

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 a square root function, it is important to apply the power rule, as the square root can be expressed as x^(1/2). Understanding how to differentiate square root functions is crucial for solving problems involving derivatives of such expressions.

27–76. Calculate the derivative of the following functions.y=10x+1y=\(\sqrt{10x+1}\)