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) f(2)=□

Explanation:

Step1: Apply sum - rule of derivatives

The sum - rule 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 at that point. For $y = f(x)$, by observing the graph of $y = f(x)$ at $x = 2$, we find the slope of the tangent line. Let's assume from the graph, the slope of the tangent line of $y=f(x)$ at $x = 2$ is $m_1$.

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

Similarly, for $y = g(x)$, by observing the graph of $y = g(x)$ at $x = 2$, we find the slope of the tangent line. Let's assume from the graph, the slope of the tangent line of $y = g(x)$ at $x = 2$ is $m_2$.

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

$F^{\prime}(2)=m_1 + m_2$.

Since the actual graph is not fully legible to determine the exact slopes $m_1$ and $m_2$, if we assume from a proper - scaled graph that $f^{\prime}(2)=a$ and $g^{\prime}(2)=b$, then $F^{\prime}(2)=a + b$.

Answer:

$F^{\prime}(2)=f^{\prime}(2)+g^{\prime}(2)$ (The actual numerical value depends on the slopes of the tangent lines of $f$ and $g$ at $x = 2$ which should be determined from the graph accurately)