Get an implicit derivative?

Questions
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1

Hi, example dy/dx x²+y²=16 is -x/y, I can’t find solution in Wolfram language book and as could be expect D[x^2 + y^2 - 16, x] is not the way.

How can get implicit derivatives? Thanks.

2

DSolve[x^2 D[y, x] + y^2 == 16, y, x] // FullSimplify

3

Solve[Dt[x^2 + y^2 == 16, x], Dt[y, x]]
(* {{Dt[y, x] → -(x/y)}} *)

Or:

Solve[Dt[x^2 + y[x]^2 == 16, x], y'[x]][[1]] /. y[x] -> y;
(* {Derivative[1][y][x] -> -(x/y)} *)
4

@ Mariusz : Coool.

Interesting:

yy = -4 Tanh[4 (1/x + C[1])]

Then

x^2 D[yy, x] + yy^2 // FullSimplify

But

D[yy, x] - (-(x/yy)) // FullSimplify

doesn’t simplify to zero

5

These are different results because they are solutions to different problems.

6

Yep, of course. I was preoccupied by the - x / y from above. The ( or better a ) correct relation is

D[yy, x, x] + (2 (x + yy) D[yy, x])/x^2 // FullSimplify