# Question #7c49e

Dec 19, 2017

$\frac{9}{20}$

#### Explanation:

the slope is given by $y ' \left(4\right)$ where $y ' \left(x\right)$ is the derivative of $y = \ln \left({x}^{2} + x\right)$ with respect to x.

$y ' \left(x\right) = \frac{d}{\mathrm{dx}} \left(\ln \left({x}^{2} + x\right)\right)$
$y ' \left(x\right) = \left(\frac{1}{{x}^{2} + x}\right) \left(\frac{d}{\mathrm{dx}} \left({x}^{2} + x\right)\right)$ (ln(x) derivative and chain rule)
$y ' \left(x\right) = \frac{1}{{x}^{2} + x} \cdot \left(2 x + 1\right)$ (power rule)

now evaluate at $x = 4$:
$y ' \left(4\right) = \left(\frac{1}{{\left(4\right)}^{2} + \left(4\right)}\right) \cdot \left(2 \left(4\right) + 1\right)$

$y ' \left(4\right) = \left(\frac{1}{20}\right) \cdot \left(9\right) = \frac{9}{20}$