00:01
Okay, so suppose i want to find the derivative by using logarithmic differentiation.
00:06
What does this mean? well, first step is i'm going to take the ln of both sides, the natural log of both sides.
00:13
I could just take log base 10, but recall that ln is log base e, which is the inverse of e to the x, so that will cancel that.
00:22
So this will be ln of y, and let's break this down by rules of logarithms.
00:27
So rules of logarithms say anything that's multiplied together can be separated by addition.
00:37
So this will be ln of 2 plus ln of x squared plus ln of the square root of 7x to the fourth plus 1.
00:47
That's all 1 because those three things are multiplied.
00:49
I see that these are multiplied as well, but i'm dividing, so i'm going to subtract ln of e to the negative 3x, and then minus as well ln of negative 3x minus 1.
01:03
Before i continue on, any exponents can go in front as well.
01:08
So this 2 could go in front as 2 ln of x.
01:11
This square root, that's the same thing as 1 half goes out front.
01:17
Ln and e will cancel, so i'll have negative 3x minus.
01:21
So simplifying this a little more, ln of y equals ln of 2 plus 2 ln of x plus 1 half ln of 7x to the fourth plus 1.
01:34
This will be minus negative 3x or plus 3x minus ln of negative 3x minus 1.
01:42
Now what i'm going to do is take the derivative.
01:45
Recall that the derivative of ln the reason why i like ln as well is because the derivative is easy it's just 1 over what's ever is inside of l and i'm going to take the derivative with respect to x now when i do this element of y it's 1 over y times dy over dx which i'm ultimately going to get by itself that's the only one i do because y is a different variable here let's take the the derivative.
02:12
Derivative of element two, just a constant, cancels...