Example 2: Find the equation of the tangent line to y = x^3 + x^2 at (3, 36).
The derivative looks like the following:
= 3x^2 + 2xSo, the slope is
= 3(3)^2 + 2(3) + 33
The equation looks like the following:
(y − 36) = 33(x − 3), or y = 33x − 63Naturally, there are a couple of things that can be done to make the problems harder. First of all, you can
be given only the x-coordinate. Second, the equation can be more difficult to differentiate.
In order to find the y-coordinate, all you have to do is plug the x-value into the equation for the curve and
solve for y. Remember this: You’ll see it again!
Example 3: Find the equation of the tangent line to y = at x = 1.
First, find the y-coordinate.
y (1) = −Second, take the derivative.
You’re probably dreading having to simplify this derivative. Don’t waste your time! Plug in x = 1 right
away.
Now, we have a slope and a point, so the equation is