00:01
So we're given a function f of x is x to the fourth power x to the fourth and we're asked to find the tangent line equation of f prime of x when x equals 3 yeah so what does this mean? so we're given this function and there's a line y equals f of x here actually i rewrite that y equals f prime of x so that's the the graph of the first derivative of the function and we have to find a tangent line equation at this point x equals 3 so what we have to do is first find the first derivative because that's what the graph is and what a tangent line means is we'll draw a quick example here so some curve the tangent line at a certain point is a straight line that gives the slope of that curve at a certain point.
01:36
So the tangent means that we have to take another derivative.
01:41
So if we just get started here, f prime of x, so the first derivative is 4x cubed.
01:52
So now we know that y equals 4x cubed.
01:55
We have to find the tangent to that at x equals 3.
01:58
So to do that, we're going to have to find the second derivative f double prime of x equals 12x squared.
02:18
So now we have the second derivative, and it's important to know that this is not the final equation because this only gives the value of the slope.
02:34
Because if you look back the definition of the derivative, if we plug in some number x, so for example, x equals 3 here, f double prime of 3 equals plug into 3 for x, we get 108...