Question

Implicit differentiation with Chain rule?
(x+y^2)^10 = 9x^2+4

Answer 1

Well when you implicitly differentiate, just keep in mind that y is a function of x, so you must chain rule y

\[(x+y^2)^{10} = 9x^2 + 4\]

now we could expand the left side, but that would take forever, so we can actually just go ahead and differentiate it in implicit form.

Whats the derivative of (x+y^2)^10? we'll have to chain rule. But remember y is a function of x, so you'll have to chain rule out a y'. The derivative of the left hand side is straight forward.

\[10(x+y^2)^9 (1+2y(y')) = 18x\]

now we need to isolate y' to finish the problem:
\[(1+2y(y')) = (18x) / 10(x+y^2)^9\]
\[2y(y')) = [(18x) / 10(x+y^2)^9] - 1\]
\[y' = (1/2y)(18x) / 10(x+y^2)^9 - 1\]

now usually its enough to leave the y in the solution, but if you want to get rid of it, you can isolate y in the original equation and plug it in.

Answer 2

Omgosh thank u so much! Idk why I was stuck on that one problem :)