# What is the equation of the line that is normal to f(x)= x/sqrt( 3x+2)  at  x=4 ?

Jan 9, 2016

Do this:
1)Find f'(x)
This is the slope of the tangent line to the graph of f(x). Let's call it m_t.
You want to find m_n, which is -1/(m_t). Recall if two lines are perpendicular, their respective slopes are negative reciprocals of one another. I've inserted a pretty TI screenshot, displaying the graph of f(x) with tangent line and normal line intersection.
What is that point of intersection?
2)Find the point (4,f(4)).
3)Use the point-slope equation for a line
y-y1=(m_n)(x-x1), where x1=4 and y1=f(4) and (m_n)=f'(4)
That's your equation for the line.

Your f'(x) should look like this....I don't want to do a step by step, cause that's your job...
$\frac{3 x + 4}{2 \cdot {\left(3 x + 2\right)}^{\frac{3}{2}}}$