SOLUTION: Hi Can somebody please help me with this problem Confirm that the point P(1,1) lies on the curve with equation x^3-y^2+xy-x^2=0 and find the values of dy/dx and d^2y/dx^2 at that

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons -> SOLUTION: Hi Can somebody please help me with this problem Confirm that the point P(1,1) lies on the curve with equation x^3-y^2+xy-x^2=0 and find the values of dy/dx and d^2y/dx^2 at that       Log On


   

Question 377490: Hi Can somebody please help me with this problem
Confirm that the point P(1,1) lies on the curve with equation x^3-y^2+xy-x^2=0 and find the values of dy/dx and d^2y/dx^2 at that point.
Please explain steps if anyone is so kind as to help
Thank you
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
x%5E3-y%5E2%2Bxy-x%5E2=0
1%5E3-1%5E2%2B%281%29%281%29-1%5E2=0
1-1%2B1-1=0
0%2B0=0
.
.
True, so 11) is on the curve.
.
.
Use implicit differentiation,
x%5E3-y%5E2%2Bxy-x%5E2=0
3x%5E2-2y%28dy%2Fdx%29%2Bx%28dy%2Fdx%29%2By-2x=0
%28-2y%2Bx%29%28dy%2Fdx%29=-3x%5E2%2B2x-y
dy%2Fdx=%28-3x%5E2%2B2x-y%29%2F%28x-2y%29
When x=y=1
dy%2Fdx=%28-3%2B2-1%29%2F%281-2%29
dy%2Fdx=%28-2%29%2F%28-1%29
highlight%28dy%2Fdx=2%29
.
.
.
For the second derivative, use the quotient rule,
r=p%2Fq
dr%2Fdx=%28q%2A%28dp%2Fdx%29-p%2A%28dq%2Fdx%29%29%2Fq%5E2
.
.
dy%2Fdx=%28-3x%5E2%2B2x-y%29%2F%28x-2y%29
.
.

The value when x=y=1 and dy%2Fdx=2 is,

d%28dy%2Fdx%29%2Fdx=%28%28-1%29%28-6%29-%28-2%29%28-3%29%29%2F%28-1%29%5E2
d%28dy%2Fdx%29%2Fdx=%286-6%29%2F1
highlight%28d%28dy%2Fdx%29%2Fdx=0%29