What is the equation of the tangent line of #f(x) =x^2-sqrt((3x^2-9x)/(x+6))# at # x = 3#?

and after that, you can use the formula to get this function's linear approximation.

To find the equation of the tangent line of f(x) = x^2 - sqrt((3x^2 - 9x)/(x + 6)) at x = 3, we need to find the derivative of f(x) and evaluate it at x = 3. The derivative of f(x) is given by:

f'(x) = 2x - (1/2) * (3x^2 - 9x)^(-1/2) * (6x - 9)/(x + 6)^2

Evaluating f'(x) at x = 3, we get:

f'(3) = 2(3) - (1/2) * (3(3)^2 - 9(3))^(-1/2) * (6(3) - 9)/(3 + 6)^2

Simplifying the expression, we find:

f'(3) = 6 - (1/2) * (27 - 27) * (9 - 9)/(9)^2

f'(3) = 6

Therefore, the slope of the tangent line at x = 3 is 6. To find the equation of the tangent line, we use the point-slope form:

y - y1 = m(x - x1)

Substituting the values x1 = 3, y1 = f(3) = 3^2 - sqrt((3(3)^2 - 9(3))/(3 + 6)), and m = 6, we have:

y - (3^2 - sqrt((3(3)^2 - 9(3))/(3 + 6))) = 6(x - 3)

Simplifying the equation, we get:

y - (9 - sqrt(0)) = 6x - 18

y - 9 = 6x - 18

y = 6x - 9

Therefore, the equation of the tangent line of f(x) = x^2 - sqrt((3x^2 - 9x)/(x + 6)) at x = 3 is y = 6x - 9.