This week was all about first and second implict derivatives and related rates. I'm actually comfortable with doing implict derivatives. The only problem I have sometimes is forgetting to put dy/dx or little mistakes with algebra. Related rates, however, I'm having difficulty with. It kind of reminds me of optimization, which I'm really really really bad at. Some of the problems that are straight forward, I can sort of do, but the others are just too complicated.
so implict derivatives:
Steps
1. take the derivative of both sides
2. everytime you take the derivative of y note it with dy/dx or y^1
3. solve for dy/dx
Example
y=sinxy
dy/dx= cosxy(x(dy/dx)+y)
dy/dx-xcosxy(dy/dx)=ycosxy
dy/dx= ycos(xy)/1-xcox(xy)
now second derivative of impliit derivatives:
It is the same steps except when you take the derivative of dy/dx, it becomes d^2y/dx^2.
Example
x^2+y^2=25 Find d^2y/dx^2
2x+2y(dy/dx)=0
dy/dx= -x/y
d^2y/dx^2= y(1)-[x(dy/dx)]/y^2
d^2y/dx^2= -y-x(-x/y)/y^2
=((y^2/y)+(x^2/y))/y^2
d^2y/dx^2= (-y^2+x^2)/y^3
Ralated Rates are waht I'm really having trouble with. Finding the equations is what trips me up. Does anyone have any advice on how to attack these problems? haha
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You first have to identify what you are Given (it helps to right them down). Then determine what you are looking for (what rate?). Sketch out what you have. Now the hard part: Write an equation involving what you have (for example: if the problem is talking about the rate at which an area is expanding: use the formula for the area being talked about). Next, take the derivative of this equation and solve for the unknown rate.
ReplyDeleteWhat equation to use depends on what the problem is telling you or asking you to find. The first example in our notes says what rate is the total AREA a of the disturbed water changing. Since they are asking for area, you know you will have to use the area formula.
ReplyDeleteExample two says air is being pumped into a spherical balloon, because of that you know you have to use volume formula of a sphere because it says spherical balloon.
I guess the only help i can really give is to read the problem carefully because the formula needed is stated in the problem.