`f(x) = 5/(x^3 - 2), (-2,(-1/2))` Find and evaluate the derivative of the function at the given point. Use a graphing utility to verify your result.

Expert Answers
justaguide eNotes educator| Certified Educator

The function f(x) = 5/(x^3 - 2) = 5*(x^3 - 2)^-1.

The derivative f'(x) = 5*(x^3 - 2)^-2*-1*3x^2

= (-15x^2)/(x^3 - 2)^2

At x = -2, f'(x) = (-15*4)/(-8 - 2)^2 = -60/100 = -0.6

The equation of the tangent to the curve f(x) = 5/(x^3 - 2) at (-2, -1/2) is:

(y + 0.5)/(x +2) = -0.6

y+ 0.5 = -0.6*x - 1.2

y = -0.6*x - 1.7

The following graph shows the function f(x) = 5/(x^3 - 2) and the line y = -0.6*x - 1.7. Note that it is tangent to the curve at the given point, verifying the derivative.