Given that f'(3) = 6 and g'(3) = -2 find (f+g)'(3).

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Step 1: Understand the problem. We are given the derivatives of two functions, f and g, at x = 3. Specifically, f'(3) = 6 and g'(3) = -2. We need to find the derivative of the sum of these functions, (f+g)'(3).

Step 2: Recall the rule for the derivative of a sum. The derivative of the sum of two functions is the sum of their derivatives. Mathematically, this is expressed as (f+g)'(x) = f'(x) + g'(x).

Step 3: Apply the rule to the given point. Substitute x = 3 into the formula: (f+g)'(3) = f'(3) + g'(3).

Step 4: Substitute the given values. We know f'(3) = 6 and g'(3) = -2, so substitute these values into the equation: (f+g)'(3) = 6 + (-2).

Step 5: Simplify the expression. Combine the values to find the derivative of the sum at x = 3.

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주요 개념

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

Derivative of a Function

The derivative of a function at a point measures the rate at which the function's value changes as its input changes. It is denoted as f'(x) and represents the slope of the tangent line to the function's graph at that point. Understanding derivatives is crucial for analyzing how functions behave locally.

Sum Rule of Derivatives

The Sum Rule states that the derivative of the sum of two functions is equal to the sum of their derivatives. Mathematically, if f and g are functions, then (f + g)'(x) = f'(x) + g'(x). This rule simplifies the process of finding derivatives when dealing with combined functions.

Evaluating Derivatives at a Point

To evaluate the derivative of a function at a specific point, you substitute the point's value into the derivative function. In this case, knowing f'(3) and g'(3) allows us to find (f + g)'(3) by directly applying the Sum Rule and substituting the given values.