What is the equation of the tangent line of f(x)=-1/x at x=3?

1 Answer
Jan 6, 2016

The equation of the tangent line is: 9y + 6 = x.

Explanation:

By the power rule, we can find f'(x) which is the equation for the 'slope' or the derivative of f(x).

f(x) = - x^(-1)

f'(x) = (-1) * (-1) x^(-2)

f'(x) = 1/x^2

Evaluating at the given x value,

f'(3) = 1/3^2

f'(3) = 1/9

The point that we're interested in is (3, -1/3), where I simply evaluated f(3) to find the y-coordinate.

So, we know that the tangent line goes through the point (3, -1/3) with a gradient of m = 1/9.

Thus, we can find the equation of that line through the 'point-slope' method.

y - (-1/3) = 1/9(x - 3)

y = 1/9 x - 3/9 - 1/3

y = 1/9 x - 2/3

Which can also be written as 9y = x - 6.