Find the derivative function f' for the following functions f.
f(x) = √3x+1; a=8

검증된 단계별 안내

Step 1: Recognize that the function f(x) = \(\sqrt{3x+1}\) is a composition of functions, specifically the square root function and a linear function. This suggests the use of the chain rule for differentiation.

Step 2: Identify the outer function and the inner function. Here, the outer function is g(u) = \(\sqrt{u}\) and the inner function is h(x) = 3x + 1.

Step 3: Differentiate the outer function g(u) with respect to u. The derivative of \(\sqrt{u}\) is \(\frac{1}{2\sqrt{u}\)}.

Step 4: Differentiate the inner function h(x) with respect to x. The derivative of 3x + 1 is 3.

Step 5: Apply the chain rule, which states that the derivative of a composite function f(x) = g(h(x)) is f'(x) = g'(h(x)) \(\cdot\) h'(x). Substitute the derivatives found in steps 3 and 4 into this formula to find f'(x).

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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. The derivative is often denoted as f'(x) and provides critical information about the function's behavior, such as its slope at any given point.

Power Rule

The Power Rule is a fundamental technique for finding derivatives, particularly for polynomial functions. It states that if f(x) = x^n, where n is a real number, then the derivative f'(x) = n*x^(n-1). This rule simplifies the differentiation process, allowing for quick calculations of derivatives for functions involving powers of x.

Chain Rule

The Chain Rule is a method for differentiating composite functions. If a function can be expressed as f(g(x)), where g(x) is another function, the Chain Rule states that the derivative f'(x) = f'(g(x)) * g'(x). This rule is essential when dealing with functions that are nested within one another, allowing for the correct application of differentiation.

Find the derivative function f' for the following functions f.f(x) = √3x+1; a=8