Derivatives and tangent lines
a. For the following functions and values of a, find f′(a).
f(x) = √2x+1; a= 4

검증된 단계별 안내

Step 1: Identify the function f(x) = \(\sqrt{2x + 1}\) and the point a = 4 where we need to find the derivative f'(a).

Step 2: Use the chain rule to differentiate f(x). The chain rule states that if you have a composite function f(g(x)), then the derivative is f'(g(x)) * g'(x).

Step 3: Recognize that f(x) = (2x + 1)^{1/2}. Differentiate the outer function (2x + 1)^{1/2} with respect to (2x + 1), which gives (1/2)(2x + 1)^{-1/2}.

Step 4: Differentiate the inner function 2x + 1 with respect to x, which gives 2.

Step 5: Combine the results from Steps 3 and 4 using the chain rule: f'(x) = (1/2)(2x + 1)^{-1/2} * 2. Simplify this expression and then substitute x = 4 to find f'(4).

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주요 개념

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

Derivatives

A derivative represents the rate of change of a function with respect to its variable. It is defined as the limit of the average rate of change of the function as the interval approaches zero. In practical terms, the derivative at a point gives the slope of the tangent line to the function at that point.

Tangent Lines

A tangent line to a curve at a given point is a straight line that touches the curve at that point without crossing it. The slope of the tangent line is equal to the derivative of the function at that point. This concept is crucial for understanding how functions behave locally around specific values.

Function Evaluation

Function evaluation involves substituting a specific value into a function to determine its output. In the context of derivatives, evaluating the function at a point helps in calculating the derivative at that point. For example, to find f′(a), one must first evaluate f(a) and then apply the derivative rules.

Derivatives and tangent linesa. For the following functions and values of a, find f′(a).f(x) = √2x+1; a= 4