# Another graph involving differentiation. This time I got j'(2) and j'(3) right, but I need help finding j'(1). Can someone help please?

Feb 3, 2018

$- \frac{4}{3}$

#### Explanation:

This question is implicitly attempting to test you on the quotient rule.

We're given that $j \left(x\right) = g \frac{x}{f} \left(x\right)$. Hence:

$j ' \left(x\right) = \frac{g ' \left(x\right) \cdot f \left(x\right) - f ' \left(x\right) \cdot g \left(x\right)}{f \left(x\right)} ^ 2$

This is just applying the quotient rule formula.

Now, we're being asked to find $j ' \left(1\right)$. Now let's look at the graph, and list everything we know about $f \left(x\right)$ and $g \left(x\right)$ at $x = 1$:

$f \left(1\right) = g \left(1\right) = \frac{3}{2}$
We can draw this relationship since f(x) and g(x) intersect at x = 1.

$f ' \left(1\right) = \frac{3}{2}$
$g ' \left(1\right) = - \frac{1}{2}$
Note that since both f(x) and g(x) are linear functions, we can simply use the slope formula to find their derivatives.

Now, we plug all this into the formula:

j'(1) = [g'(1)xxf(1) - f'(1)xxg(1)]/[f(1)]^2 = [(-1/2) xx (3/2) - (3/2)xx(3/2)]/[(3/2)^2]
$= - \frac{4}{3}$

If you need some additional help/practice on using the quotient rule, I'd encourage you to take a look at some of my videos on the subject:

Hope that helps :)