Find the slope of the graph of the function at the given point. Use the derivative feature of a graphing utility to confirm your results. (If an answer is undefined, enter UNDEFINED.)

Question

y =

x2 + 9x,

(3, 6)

Answer 1

y=sq(x^2+9x) = (x^2+9x)^(1/2) through the point (3,6)

6=sqr(3^2+9(3)) = sqr36 = 6

y' = (1/2)(2x+9)/sqr(x^2+9)= (x+4.5)/sqr(x^2+9)

y'(3) = (3+4.5)/sqr(3^2+9) = 7.5/sqr(18) = 7.5(3sqr2)/18

slope at the point (3,6) = 22.5sqr2/18 = 1.25sqr2

= about 707/400 = 1.7675

y=sqr(x^2+9x)

y^2 = x^2+9x

y^2-x^2 +9x = 0

y^2 -(x^2 - 9x+ 81/4) = -81/4

-4y^2/81 + (4/81)(x-9/2)^2 = 1

(x-9/2)^2/(9/2)^2 - y^2/(9/2)^2 = 1

is a horizonta hyperbola, but ignore the bottom half of the branches as y cannot be less than zero

Answer 2

1. write f(x) as (x²+9)^1/2
2. Use power rule followed by chain rule to f;(x)
3. 1/2 (x²+9)^-1/2(2x+9)
4. Find f'(3) to determine the slope at that point

Check your work here:

desmos.com/calculator/fteffmhiq8