`f(x) = x/(x+4), (-5,5)` (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...

`f(x) = x/(x+4), (-5,5)` (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.

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Textbook Question

Chapter 2, 2.3 - Problem 65 - Calculus of a Single Variable (10th Edition, Ron Larson).
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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to evaluate the equation of the tangent line at (-5,5), using the formula:

`f(x) - f(-5) = f'(-5)(x + 5)`

Notice that f(-5) = 5.

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

`f'(x) =((x)'(x+4) - (x)(x+4)')/((x+4)^2)`

`f'(x) = (x+4 - x)/((x + 4)^2)`

`f'(x) = 4/((x+4)^2)`

You need to evaluate the derivative at x = -5:

`f'(-5) = 4/((-5+4)^2) =>< f'(-5) = 4`

Replacing the values into equation yields:

`f(x) - 5= 4(x + 5)`

`f(x) = 4x + 25`

Hence, evaluating the equation of the tangent line at the given curve, yields `f(x) = 4x + 25.`

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