derivative of arctan(y/x)
We’re going to find the first partial derivatives of this function and evaluate them at 4, negative for let’s go ahead and go through it carefully before we do the derivative of arctan(y/x) for the call that the derivative of arctangent is 1 / 1 + x squared so if you just had DDX of arctangent say simply be well be 1/1 + x sqrt so we’re going to use this formula and this problem so first of time bail after sex so taxes owed LF du lac’s is the partial derivative with respect to X we do this all of the wise are constants.
Now we have an X in here so this is not a concert so we still have to use this formula so it’s going to be one 1/1 + one over one plus x squared but this is RX this whole thing so y over X squared times the derivative of the inside function so we have this piece with respect to X keeping in mind that why is a constant so what I’m thinking is we can think of Y over X eye x 1 / X which is why y x x times x to the negative 1 one and why is a constant.
when you differentiate this is putting negative in you say tract you subtract one right you just differentiate the X and Y & Y without cuz it’s a constant and I think I’m going to write it like this so this is going to be negative y over x squared over / 1 + y squared over x squared by squaring each piece then bring this down in writing it like outside that s basically wrote that here this is up top and then.
We have that on the bottom so now what we can all we can do is we can * x squared over x squared to clean things up to do the t do that that please cancel so we get Negative why is really beautiful export * 1 is simply t time e is it that cancels so you get white square to Waterford answer so that my friends would be dull F 2X Dell Optiplex be driven of a bath with respect to X 2 slope in the X Direction x-direction. why so tell f F. Why all this would be 1 / same thing 1 + y over X quantity squared times the derivative of the inside function so now we’re taking the derivative with respect to y Sol so I can’t you can think of it like this
The derivative of y is one the one over X hangs out so you just get x 1 / X that’s all you would get there I believe yep yep that’s it because like the ore like before the derivative of wise one and this just hangs out I’m going to write it like this one over x / 1 + y squared over x squared and I’m leaning towards doing something similar so let’s go ahead and put (here in) here hear the same thing as x x squared over x squared just to make it look better.
I mean to do that by we can do that by the way we’re basically multiplying by 1/4 I do this you lose Annex here so you get EX and then here export * 1 is x squared x squared times this to get a y squared really really cool really pretty looking derivatives difference at this point this is your ex this is your way so let’s do the LF The Lacs at 4 four negative four for a negative for that would be this your wires negative form is already negative here so it’ll be a positive for on the bottom you get for his purpose for Aquarius so you get 16 + 16 16 + 16 + 4/32 before goes into for one time.
32 so it’ll be 1/8 so this would be could be the slope and the extra action at that point now it’s fine in the other one at the same point so dull F the Y at four negative four happened on this problem so it’s my first time doing it I didn’t like we rehearsed the problem breathing I probably work this out about a year ago so it’s interesting we got 1/8 in an interesting a how pretty the answers look okay so this is our four and then same thing 16 + 16 16 + 16 really nice so this case we also get for 4/32 o that’s really cool we got the same exact answer kind of nice right kind of nice I to hope this video has been helpful to anyone out there who is learning some stuff with derivative of arctan(y/x).