Question

For `f(x)=1/x` what is the equation of the tangent line at `x=1/2`?

Answer 1

`y=-4x+4`

Explanation:

We must find `f'(1/2)` to determine the slope of the tangent line.

Remember that the derivative of `x^n` is `nx^(n-1)`.

`f(x)=x^-1`
`f'(x)=-x^(-2)`
`f'(x)=-1/x^2`

`f'(1/2)=-1/(1/2)^2=-1/(1/4)=-4`

The slope of our tangent line at `x=1/2` is `-4`. We know that the point the tangent line touches is `(1/2,2)` by finding `f(1/2)`.

We can use point-slope form to write our linear equation:
`y-2=-4(x-1/2)`

If you want your answer in slope-intercept form:
`y=-4x+4`