Given that f(1)=2 and f′(1)=2 , find the slope of the curve y=xf(x) at the point (1, 2).

검증된 단계별 안내

Step 1: Identify the function y = x f(x) and recognize that you need to find the derivative of this function to determine the slope of the curve at a specific point.

Step 2: Use the product rule to differentiate y = x f(x). The product rule states that if you have two functions u(x) and v(x), then the derivative of their product is given by (uv)' = u'v + uv'.

Step 3: Apply the product rule to y = x f(x). Here, u(x) = x and v(x) = f(x). Therefore, the derivative y' = (x)'f(x) + x(f(x))'.

Step 4: Calculate the derivatives: (x)' = 1 and (f(x))' = f'(x). Substitute these into the expression from Step 3 to get y' = 1*f(x) + x*f'(x).

Step 5: Substitute the given values f(1) = 2 and f'(1) = 2 into the derivative expression y' = f(x) + x*f'(x) to find the slope at the point (1, 2).

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주요 개념

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

Product Rule

The Product Rule is a fundamental differentiation technique used when finding the derivative of a product of two functions. It states that if you have two functions u(x) and v(x), the derivative of their product is given by u'v + uv'. This rule is essential for differentiating the function y = x f(x) in the given problem.

Derivative

A derivative represents the rate of change of a function with respect to its variable. It provides information about the slope of the tangent line to the curve at any given point. In this context, calculating the derivative of y = x f(x) will allow us to find the slope of the curve at the specific point (1, 2).

Evaluating Derivatives at a Point

Evaluating derivatives at a specific point involves substituting the x-value of that point into the derivative function. This process yields the slope of the tangent line to the curve at that point. In this case, after applying the Product Rule and finding the derivative, we will substitute x = 1 to determine the slope of the curve y = x f(x) at (1, 2).