SOLUTION: implicit diffrention problem Use implicit differentiation to find an equation of the tangent line to the curve at the given point. x^2 + y^2 = (2x^2 + 5y^2 − x)^2 (0,1/5)

Algebra ->  Functions -> SOLUTION: implicit diffrention problem Use implicit differentiation to find an equation of the tangent line to the curve at the given point. x^2 + y^2 = (2x^2 + 5y^2 − x)^2 (0,1/5)       Log On


   

Question 1154662: implicit diffrention problem
Use implicit differentiation to find an equation of the tangent line to the curve at the given point.
x^2 + y^2 = (2x^2 + 5y^2 − x)^2
(0,1/5)



Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
Since "derivative at a point" means "slope of the tangent line at that point',
this problem could be stated this way:

Find the equation of the line through
%28matrix%281%2C3%2C0%2C%22%2C%22%2C1%2F5%29%29, 
which has its slope equal to the
derivative of

x%5E2+%2B+y%5E2+=+%282x%5E2+%2B+5y%5E2+-+x%29%5E2

at that point.

Take derivatives term by term:

2x%2B2y%2Aexpr%28dy%2Fdx%29=2%282x%5E2%2B5y%5E2-x%29%284x%2B10y%2Aexpr%28dy%2Fdx%29-1%29

Divide through by 2

x%2By%2Aexpr%28dy%2Fdx%29=%282x%5E2%2B5y%5E2-x%29%284x%2B10y%2Aexpr%28dy%2Fdx%29-1%29

Substitute x=0



Simplify

y%2Aexpr%28dy%2Fdx%29=%285y%5E2%29%2810y%2Aexpr%28dy%2Fdx%29-1%29

Substitute y=1/5



Simplify

%281%2F5%29%2Aexpr%28dy%2Fdx%29=%285%281%2F25%29%29%282%2Aexpr%28dy%2Fdx%29-1%29

%281%2F5%29%2Aexpr%28dy%2Fdx%29=%281%2F5%29%282%2Aexpr%28dy%2Fdx%29-1%29

Multiply through by 5

expr%28dy%2Fdx%29=2%2Aexpr%28dy%2Fdx%29-1

-expr%28dy%2Fdx%29=-1

expr%28dy%2Fdx%29=1

So since the slope of the tangent line is the derivative
at the point 
%28matrix%281%2C3%2C0%2C%22%2C%22%2C1%2F5%29%29, 
we want the equation of the line through that point with slope m = 1.

y-y%5B1%5D=m%28x-x%5B1%5D%29
y-1%2F5=1%28x-0%29
y-1%2F5=x
y=x%2B1%2F5 

That's the answer.  Here's the graph:



Edwin