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))

Explanation:

Step1: Recall sum - rule of derivatives

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 $y = f(x)$ at $x = 2$, using the two - point formula for slope $m=\frac{y_2 - y_1}{x_2 - x_1}$. Consider two points on the tangent line of $y = f(x)$ at $x = 2$, say $(0,2)$ and $(4,0)$. Then $f^{\prime}(2)=\frac{0 - 2}{4 - 0}=-\frac{1}{2}$.

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

For $y = g(x)$ at $x = 2$, consider two points on the tangent line of $y = g(x)$ at $x = 2$, say $(0,4)$ and $(4,0)$. Then $g^{\prime}(2)=\frac{0 - 4}{4 - 0}=- 1$.

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

$F^{\prime}(2)=f^{\prime}(2)+g^{\prime}(2)=-\frac{1}{2}+(-1)=-\frac{1 + 2}{2}=-\frac{3}{2}$.

Answer:

$-\frac{3}{2}$