Question

How do you find the equation of the line tangent to `y=x^3 - 2x` at the point (2,4)?

Answer 1

`y=10x-16`

Explanation:

Step 1: Take the Derivative
Using the power rule and sum rule, we see that `y'=3x^2-2`.

Step 2: Evaluate to find the Slope
We are looking for the equation of the tangent line, and one component of that is the slope. Since the slope is the derivative at the point, we can evaluate our derivative at `x=2` to find the slope:
`y'=3(2)^2-2`
`y'=10`

Step 3: Finding the Equation
Now that we have the slope (`10`) and a point `(2,4)`, we can find the equation. Tangent lines are of the form `y=mx+b`, where `x` and `y` are points on the line, `m` is the slope, and `b` is the `y`-intercept. All we are missing is the `y`-intercept, so that's what we'll solve for:
`y=mx+b`
`4=10(2)+b`
`4=20+b`
`b=-16`

Putting all the information together, the equation of the tangent line at `(2,4)` is `y=10x-16`.