Question

let f = f + g, where the graphs of f and g are shown in the figure to the right. find the following derivative. f(2)
question completed: 18 of 18 my score: 17.6/18 pts (97.8%)

Explanation:

Step1: Apply sum - rule of derivatives

The sum - rule of derivatives states that if \(F(x)=f(x)+g(x)\), then \(F^{\prime}(x)=f^{\prime}(x)+g^{\prime}(x)\). So \(F^{\prime}(2)=f^{\prime}(2)+g^{\prime}(2)\).

Step2: Find \(f^{\prime}(2)\) from the graph

The derivative of a function at a point is the slope of the tangent line to the graph of the function at that point. For \(y = f(x)\) at \(x = 2\), using the two - point formula for slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\). If we take two points on the tangent line of \(y = f(x)\) at \(x = 2\) (say \((0,0)\) and \((4,4)\) on the tangent line), then \(f^{\prime}(2)=\frac{4 - 0}{4 - 0}=1\).

Step3: Find \(g^{\prime}(2)\) from the graph

For \(y = g(x)\) at \(x = 2\), using the two - point formula for slope. If we take two points on the tangent line of \(y = g(x)\) at \(x = 2\) (say \((0,4)\) and \((4,0)\) on the tangent line), then \(g^{\prime}(2)=\frac{0 - 4}{4 - 0}=- 1\).

Step4: Calculate \(F^{\prime}(2)\)

Substitute \(f^{\prime}(2)\) and \(g^{\prime}(2)\) into \(F^{\prime}(2)=f^{\prime}(2)+g^{\prime}(2)\). So \(F^{\prime}(2)=1+( - 1)=0\).

Answer:

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