What is the derivative of tan^-1 (xy) = 1+ x^2y?
1 Answer
Apr 17, 2016
Explanation:
We will be differentiating implicitly. On the left hand side, we will use the chain rule in regards to the inverse tangent function:
d/dx(arctan(u))=(u')/(1+u^2)
Also, note that the product rule will be used for
Differentiating gives:
(d/dx(xy))/(1+(xy)^2)=yd/dx(x^2)+x^2d/dx(y)
(yd/dx(x)+xd/dx(y))/(1+x^2y^2)=2xy+x^2dy/dx
(y+xdy/dx)/(1+x^2y^2)=2xy+x^2dy/dx
From here, just algebra your way through to an equation solved for
y+xdy/dx=(2xy+x^2dy/dx)(1+x^2y^2)
y+xdy/dx=2xy(1+x^2y^2)+dy/dx(x^2+x^4y^2)
xdy/dx-dy/dx(x^2+x^4y^2)=2xy+2x^3y^3-y
dy/dx(x-x^2-x^4y^2)=2xy+2x^3y^3-y
dy/dx=(2xy+2x^3y^3-y)/(x-x^2-x^4y^2)