Implicit Differentiation: Exercise 1

00:01 Example 2: We now have xy plus 15x equals to 2y plus 5.

00:12 A very obvious-looking implicit equation because it has a mixture of x's and y's.

00:17 We are going to differentiate this implicitly. So we've looked at it.

00:21 We've seen that there is a mixture of x's and y's.

00:24 I really don't want to go into trying to rearrange it. We have an easier method.

00:28 Let's just do this implicitly.

00:31 Okay, thinking about each term I can see here that this x times y so you must think product rule there. I can see that we have a y term here, so we're thinking dy/dx and then we've got some straightforward terms in between.

00:46 So I'm gonna make some space for my product rule here.

00:49 Remember what we said. Leave 1 term as it is, so let's leave x as it is.

00:56 Differentiate y. When you differentiate y that goes to 1 and then remember to multiply this with the factor dy/dx.

01:04 Second term, we now differentiate this, which gives us 1 and we leave the y as it is.

01:13 Easy. So we've just used the product rule and managed to differentiate x times y.

01:19 Our next term, 15x just differentiate to 15. 2y, we can see another y term.

01:27 If you were differentiating 2y with respect to y, that would just give you 2 but remember to multiply this with dy/dx.

01:35 And the 5 at the end, it's just a constant, so that goes to zero.

01:40 We're gonna tidy this equation up. So I have x times dy/dx plus y and take it out of the brackets and multiply it with a positive, so that's fine.

01:52 I have plus 15 equals to 2dy/dx, so not just 2y.

02:02 Okay, we're now going to move this often, the option is to keep things positive.

02:10 So, we're going to leave the 2dy/dx on this side. And then you can move the other 2dy/dx term here.

02:18 So my whole objective is to keep many terms positive as I can.

02:24 We've got y plus 15 on this side. You can take dy/dx as a common factor leaving you with 2 minus x and then on this side, remember you still have y plus 15 and lastly, divide by 2 minus x because it's multiplying on this side so you end up with y plus 15 over 2 minus x equals to dy/dx, which is your gradient.