Question

Find y prime if arctan(xy) = 1 + (x^2*y)

Answer 1

y prime as in dy/dx?

Answer 2

yeah derivative of the function f(x) = y

Answer 3

arctan(xy) = 1+ x^2*y => xy = tan(1+x^2y)

Differentiating w.r.t x,
y + xy' = sec^2(1+x^2y)(2x*y + y'*x^2)

Can you do the algebra on your own?

Answer 4

yeah that will definitely help, thanks

Answer 5

can you work out another step or two? i seem to have over estimated my simplifying abilities

Answer 6

If ever you have difficult deriving an inverse trigonometric function, remember that you can prove it on-spot.\[y=\arcsin x,\sin y=x,\frac{dy}{dx}\cos y=1,\frac{dy}{dx}=\sec y=\sec\left(\arcsin x\right)\]The last step can be done using a right triangle and Pythagoras' theorem.\[h=\sqrt{1-x^2},\sec\left(\arcsin x\right)=\sec y=\frac{1}{h}=\frac{1}{\sqrt{1-x^2}}\]\[\frac{dy}{dx}=\frac{1}{\sqrt{1-x^2}}\]This general sort of solution can be done with any trigonometric function.