If $ g(x) = x^4 - 2 $, find $ g'(1) $ and use it to find an equation of the tangent line to the curve
$ y = x^4 - 2 $ at the point $ (1, -1) $.
Okay here we have the function G. Of X equals exit the fourth minus two. Uh The graph of that would be a curve. And we want to find the equation of detention line To to curve to the graph of G. F. X. at the .1 Common -1. We're going to use what's called the point slope form The equation of a line. Uh that you learned back in Algebra one. Why- Why one equals M times X minus X. One. Uh This is the equation of a line. This is gonna be the equation of our tangent line X one is going to be uh And why one are going to be the coordinates of the point where line is tangent to the curb. So the line is tension to the curb at this point. The slope. Okay, so we already know why one, the next one are going to be. Uh X and Y will stay. Just excellent why they'll be part of our final equation. Uh But we have to find em and the slope of the tangent line is simply the derivative of this function G prime of X. Uh huh. Uh This X value when X is one. So what we have to do now is we have to find G prime of X. So we're gonna take the derivative of G F X with respect to X. So G prime of X is going to be the derivative of X to the fourth minus two, derivative of extra fourth is four X to the third. Uh And the derivative of a constant like to the derivative of two is zero. So we don't have to write minus zero. So that's G prime of X. So the derivative of our function G. At this X coordinate one. What is G prime of one? While G prime of X is four times X cubed. So G prime of one is four times one cube One Cube is one times 4 is four. There is going to be our slope. So let's draw some lines in here. X wine is going to go here. The Y one coordinate is going to go here and derivative. Uh that we got is going to be the slope of attention like and so our final answer will be arranged algebraic lead to clean it up a little bit. But why minus? Why one? Why one is negative one? So why subtract negative one is why plus one And that equals m. Which is four times x minus X. One. X Subtract x one X one was one. So extract one. Let's distribute this multiplication four times x minus one. We have Y plus one equals Uh four times x -1 is 4, -4. Last but not least. Let's get why by itself, by subtracting this one from both sides, and we get why equals four x minus five. So this is the equation of our tangent line, the line that is tangent to the curve, the graph of G. Fx at the 0.1 common negative one.