Derivative of the implicit Function:
arctan(x^5y)=xy^5
Here is where I am at. I am pretty frustrated:
arctan(x^5y)-xy^5=0
1/1+(x^5y)^2 * (d/dx(x^5)*y+x^5*d/dx(y))-(d/dx(x)*y^5+x*d/dx(y^5))=0
1/(1+x^5y)^2 * (5x^4y+x^5dy/dx) - (y^5+x^5y^4dy/dx)
(5x^4y)/[(1+x^5y)^2] + (x^5dy/dx)/[(1+x^5y)^2] - y^5+5y^4dy/dx=0
I move the parts of the problem that lack the dy/dx to the other side of the equation for
[X^5 (dy/dx)]/[(1+x^5y)^2] + x5y^4(dy/dx) = y^5-(5x^4y)/[(1+x^5y)^2]
I am stuck here. Should I be removing the denominator before removing the dy/dx and if so, that would also remove it on the other side of the equation right? I have tried to solve this problem 13 times. I am pretty frustrated.
I find it a lot more readable just to use y', instead of all those dy/dx. And, there's no reason to rearrange the equation.
arctan(x^5y)=xy^5
1/(1+x^10y^2) (5x^4 + x^5y') = y^5 + 5xy^4y'
5x^4y/(1+x^10y^2) + x^5y'/(1+x^10y^2) = y^5 + 5xy^4y'
Now collect the y' stuff
(x^5/(1+x^10y^2) - 5xy^4)y' = y^5 - 5x^4y/(1+x^10y^2)
Clear the fractions. You can do this without worry, because you know the denominator is never zero.
(x^5 - 5xy^4(1+x^10y^2))y' = (y^5(1+x^10y^2) - 5x^4)
y' = (y^5(1+x^10y^2) - 5x^4y)/(x^5 - 5xy^4(1+x^10y^2))
and you can expand those polynomials if you want. No sweat, just a lot of algebra I.
I understand that you are working on finding the derivative of the given implicit function arctan(x^5y) = xy^5. You have made progress up to the point:
(5x^4y)/[(1+x^5y)^2] + (x^5dy/dx)/[(1+x^5y)^2] - y^5+5y^4dy/dx=0
To continue solving for dy/dx, you can start by multiplying both sides of the equation by [(1+x^5y)^2] to eliminate the denominators:
[(1+x^5y)^2] * [(5x^4y)/[(1+x^5y)^2] + (x^5dy/dx)/[(1+x^5y)^2] - y^5+5y^4dy/dx] = [(1+x^5y)^2] * 0
This simplifies to:
5x^4y + x^5(dy/dx) - y^5(1+x^5y)^2 + 5y^4(dy/dx)(1+x^5y)^2 = 0
Now, you can distribute the terms on both sides:
5x^4y + x^5(dy/dx) - y^5 - x^5y^6 - 2x^5y^5(dy/dx) + 5x^5y^4(dy/dx) = 0
To solve for dy/dx, gather the terms with dy/dx on one side and the remaining terms on the other side:
x^5(dy/dx) - 2x^5y^5(dy/dx) + 5x^5y^4(dy/dx) = y^5 - 5x^4y - 5x^5y^6
Factor out dy/dx:
(dy/dx)(x^5 - 2x^5y^5 + 5x^5y^4) = y^5 - 5x^4y - 5x^5y^6
Finally, divide both sides by (x^5 - 2x^5y^5 + 5x^5y^4) to solve for dy/dx:
dy/dx = (y^5 - 5x^4y - 5x^5y^6) / (x^5 - 2x^5y^5 + 5x^5y^4)
That is the derivative of the given implicit function with respect to x, dy/dx.