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so I've got a ten-foot lad ladder that's leaning against a wall but it's on very slick ground and it starts to slide outward and right when it's and right at the moment that we're looking at this ladder the base of the ladder is eight feet away from the base of the wall and it's sliding outward at four feet per second and we'll assume that the top of the ladder kind of glides along the side of the wall it stays kind of in contact with the wall and moves straight down and we see right over here the arrow is moving straight down and our question is how fast is it moving straight down at that moment so let's think about this a little bit what do we know and what do we not know so if we call the distance between let's call the district in the base of the lot the base of the wall on the base of the ladder let's call that X we know right now X is equal to eight feet we also know the rate at which X is changing with respect to time the rate at which X is changing with respect to time is four feet per second so we could call this DX DT now let's call let's call the distance between the top of the ladder and the base of the ladder H let's call that H so what we're really trying to figure out is what D H is what D H DT is given that we know all of this other information so let's see if we can come up with a relationship between X and H and then take the derivative with respect to time maybe using the chain rule and see if we can solve for D h dt knowing all of this information well we know the relationship between X and H at any time because of the Pythagorean theorem we can assume this is a right angle so we know that x squared we know that x squared plus h squared x squared plus h squared is going to be equal to the length of the ladder squared is going to be equal to 100 and what we care about is the rate at which these things change with respect to time so let's take the derivative with respect to time of both sides of this so we're doing a little bit of implicit a little bit of implicit differentiation so what's the derivative with respect to time of x squared well the derivative of x squared with respect to X is 2x and we're gonna have to multiply that times the derivative of X with respect to T dx/dt just to be clear this is a chain rule this is the derivative of x squared with respect to X which is 2x times dx/dt to get the derivative of x squared with respect to time just the chain rule now similarly what's the derivative of a squared with respect to time well that's just going to be 2h the derivative of H squared with respect to H is 2 H times the derivative of H with respect to time once again this right over here is the derivative of H squared with respect to H times the derivative of H with respect to time which gives us the derivative of H squared with respect to time and what do we get on the right-hand side of our equation well the length of our ladder isn't changing this hundred isn't going to change with respect to time the derivative of a constant is just equal zero so now we have it a relationship between the rate of change of H with respect to time the rate of change of X with respect to time and then for at a given point in time when the length of X is X and H is H but do we know what H is when X is equal to 8 feet well we can figure it out when X is equal to 8 feet we can use the Pythagorean theorem again we get 8 feet squared plus h squared plus h squared is going to be equal to 100 so 8 squared is 64 subtract it from both sides you get h squared is equal to 36 take the positive square root a negative square root doesn't make sense because then the ladder would be below the ground it would be somehow underground so we get H is equal to 6 so this is something that was essentially given by the problem so now we know we can look at this original thing right over here we know what X is that was given right now X is 8 feet we know the rate of change of X with respect to time it's 4 feet per second we know what H is right now it is 6 so then we can solve for the rate of H with respect to time so let's do that so we get 2 times 2 times 8 feet times 4 feet per second 4 feet per second so times 4 feet per second 4 feet per second so times four plus plus 2h so plus 2h is because it's going to be plus two times our height right now is six times the rate at which our height is changing with respect to T is equal to zero and so we get 2 times 8 times 4 is 64 plus 12 dhdt is equal to zero we can subtract 64 from both sides we get 12 the 12 times the derivative of H with respect to time is equal to negative 64 negative 64 and then we just have to divide both sides by 12 and so now we get a little bit of a drumroll the derivative the rate of change of H with respect to time is equal to negative 64 divided by 12 is equal to negative 64 over 12 which is the same thing as negative 16 over 3 yep that's right which is equal to let me scroll over to the right a little bit negative 5 and one-third feet per second feet per second so we're done but let's just do a reality check does that make sense that we got a negative value right over here well our height is decreasing our height is decreasing so it completely makes sense that it's rate of change is indeed that it's rate of change is indeed negative and we're done