Calculus Problem!!

Question

Find an equation for the line tangent to the graph of f(x)=4x/x+8

at the point (2,4/5).

Answer 1

The slope of the tangent line is given by the derivative.

For:

to find the derivative function, you must use the quotient rule, (u/v)' = (u'v - uv')/v² where u = 4x and v = x + 8 so u' = 4 and v' = 1 and v² = (x + 8)² making f '(x) = (4(x + 8) - (4x)(1))/(x + 8)² = (4x + 32 - 4x)/(x + 8)² = 32/(x + 8)²

So if f '(x) = 32/(x + 8)² then f '(2) = 32/100 = 0.32 (or 8/25)

Now just use the point-slope form of a line y - y1 = m(x - x1) where m = 0.32 and (x1, y1) = (2, 0.8):

y - 0,8 = 0.32(x - 2)

y = 0.32x - 0.16 or, using fractions, y = (8/25)x + 4/25