[Math] Use graph of function to find derivatives (product and quotient rule)

calculusderivativesfunctions

The graphs of the function $F$ (left, in blue) and $G$ (right, in red) are below. Let $P(x)=F(x)G(x)$ and $Q(x)=F(x)/G(x)$.

Answer the following questions.

  1. $P′(1)=$
  2. $Q′(1)=$
  3. $P′(6)=$
  4. $Q′(6)=$

Here is the graph of $y=F(x):$

enter image description here

And here is the graph of $y=G(x):$

enter image description here

I know this is asking me to be able to use the Product and Quotient rule to find the derivatives; Product rule if asking for $P'(x)$ and Quotient for $Q'(x)$. I am fairly okay with doing these two. However, I am unsure how to get the numbers I need from the graph: the $f(x), g(x)$, $f'(x)$ and $g'(x)$. I believe $f(x)$ would be $1$ and $g(x)$ would be $3$, but I am unsure–and definitely unsure about how to get the $f'(x)$ and $g'(x)$. Any help would be appreciated.

Best Answer

Since you are asked for $P'(x)$ at both $1$ and $6$, and the same for $Q'(x)$, you need to be more clear as to the values of $x$. Here is what I read straight from the graphs:

$$F(1)=1, \quad F(6)=4$$ $$G(1)=3, \quad G(6)=4$$

The derivatives are only slightly more tricky: look at the slope of the tangent line.

$$F'(1)=0, \quad F'(6)=\frac 12$$ $$G'(1)=1, \quad G'(6)=3$$

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