find dy/dx in terms of x and y if arctan(x^6y)=xy^6
-(x^12y^8+y^6-6x^5y)/(6x^13y^7+6xy^5-x^6)
To find dy/dx in terms of x and y, we can employ implicit differentiation. To do this, we differentiate both sides of the equation with respect to x, treating y as a function of x.
Let's differentiate each term step by step:
On the left side of the equation, we have arctan(x^6y). The derivative of arctan(u), where u is a function of x, is given by du/dx / (1 + u^2). Applying this rule, we get:
d/dx (arctan(x^6y)) = (d/dx (x^6y)) / (1 + (x^6y)^2)
Next, we differentiate the right side of the equation, xy^6, using product rule:
d/dx (xy^6) = x * (d/dx (y^6)) + y^6 * (d/dx (x))
= x * (6y^5 * (dy/dx)) + y^6
Now, equating the derivatives obtained from both sides of the equation:
(d/dx (x^6y)) / (1 + (x^6y)^2) = x * (6y^5 * (dy/dx)) + y^6
To find dy/dx, we need to isolate it on one side of the equation. Rearranging the terms, we obtain:
(d/dx (x^6y)) / (1 + (x^6y)^2) - x * (6y^5 * (dy/dx)) = y^6
To facilitate this further, let's multiply both sides by (1 + (x^6y)^2):
d/dx (x^6y) - x * (6y^5 * (dy/dx)) * (1 + (x^6y)^2) = y^6 * (1 + (x^6y)^2)
Expanding the right side of the equation:
d/dx (x^6y) - x * (6y^5 * (dy/dx)) - (x * (6y^5 * (dy/dx)) * (x^6y)^2) = y^6 + (x^6y * y^6)
Simplifying:
d/dx (x^6y) - x * (6y^5 * (dy/dx)) - x^13 * y^7 * (6y^5 * (dy/dx)) = y^6 + (x^6 * y^7)
Now, let's collect the terms containing dy/dx on one side:
d/dx (x^6y) - x * (6y^5 * (dy/dx)) - x^13 * y^7 * (6y^5 * (dy/dx)) + x * (6y^5 * (dy/dx)) = y^6 + (x^6 * y^7)
Combining the terms:
d/dx (x^6y) - x^13 * y^7 * (6y^5 * (dy/dx)) + x * (6y^5 * (dy/dx)) - x * (6y^5 * (dy/dx)) = y^6 + (x^6 * y^7)
The terms "- x * (6y^5 * (dy/dx))" and "+ x * (6y^5 * (dy/dx))" cancel each other out:
d/dx (x^6y) - x^13 * y^7 * (6y^5 * (dy/dx)) = y^6 + (x^6 * y^7)
Finally, let's isolate (dy/dx) by factoring it out:
(6y^5) * (dy/dx) * (x * x^13 * y^7 - 1) = y^6 + (x^6 * y^7) - d/dx (x^6y)
Simplifying:
(6y^5) * (dy/dx) * (x^14 * y^7 - x) = y^6 + (x^6 * y^7) - d/dx (x^6y)
Finally, we can write dy/dx in terms of x and y as:
dy/dx = (y^6 + (x^6 * y^7) - d/dx (x^6y)) / ((6y^5) * (x^14 * y^7 - x))
Therefore, dy/dx in terms of x and y, based on the given equation arctan(x^6y) = xy^6, is given by:
dy/dx = (y^6 + (x^6 * y^7) - d/dx (x^6y)) / ((6y^5) * (x^14 * y^7 - x))