Question

What is the equation of the tangent line of `f(x)=x^2 - 2x` at `x=5`?

Answer 1

`2y = x + 25`

Explanation:

So at that point on the curve, as it is a tangent the gradient will be the same.
The gradient is calculated by differentiating.
`dy/dx x^2 - 2x dx = 1/2x - 2`
We can then substitute `x` in to find the gradient at `x=5`.
Therefore, `1/2(5) - 2 = 1/2`.
This is our gradient.
Then we need to find `y` at the point `x = 5`.
Substitute into `x^2 - 2x` to get `(5)^2 -2(5) = 15`.
Now we can use the equation of `y - y_1 = m(x - x_1)`.
Therefore:
`y - 15 = 1/2(x - 5)`
Expand and rearrange:
`y - 15 = 1/2x - 5/2`
`y = 1/2x + 25/2`
`2y = x + 25`.