00:01
All right, so we're going to go ahead and compute the slope of the tangent line of this curve p of x and the instantaneous rate of change of this function.
00:09
So there's a weird little notation in the question, this little asterisk.
00:13
I'm going to assume it meant to put e to the 5x or, and it could be anything, but i'm just going to assume to you the 5x because it's an interesting question in this case if we just do this, because this requires the product rule.
00:22
So by taking the derivative of this function, which is what we want in terms of finding the slope of the tangent line, slope the tangent line at a point is the derivative at a particular point.
00:32
So this derivative is going to be by the product rule the first times the derivative the second, plus the derivative of the first times the second.
00:48
Great.
00:49
Let's compute these derivatives.
00:51
X cubed.
00:51
This is a chain rule, so it's e to the 5x times 5, plus 3x squared by power rule times e to the 5x.
00:59
We care about the value of this function, the value of the slope at the point x equals 3.
01:08
But maybe before we do that, we simplify a little bit.
01:11
Pull out and you do the 5x and you get 5x cubed plus 3x squared...