# `f(x) = (x+3)/(x-3), (4,7)` (a) Find an equation of the tangent line to the graph of f at the given point, (b) use a graphing utility to graph the function and its tangent line at the point,...

`f(x) = (x+3)/(x-3), (4,7)` (a) Find an equation of the tangent line to the graph of f at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of a graphing utility to confirm your results.

Asked on by enotes

### 1 Answer

sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted on

You need to evaluate the equation of the tangent line at (4,7), using the formula:

`f(x) - f(4) = f'(4)(x - 4)`

Notice that f(4) = 7.

You need to evaluate f'(x), using the quotient rule, such that:

`f'(x) =((x+3)'(x - 3) - (x + 3)(x - 3)')/((x - 3)^2)`

`f'(x) = (x - 3 - x - 3)/((x - 3)^2)`

`f'(x) = -6/((x - 3)^2)`

You need to evaluate the derivative at x = 4:

`f'(4) = -6/((4-3)^2) =>< f'(4) = -6`

Replacing the values into equation yields:

`f(x) - 7= -6(x - 4)`

f'(x) = -6x + 31

Hence, evaluating the equation of the tangent line at the given curve, yields `f'(x) = -6x + 31.`