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.
D[x^2 + y^2 - 16, x]
How can get implicit derivatives? Thanks.
@ 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
These are different results because they are solutions to different problems.
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
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)} *)
DSolve[x^2 D[y[x], x] + y[x]^2 == 16, y[x], x] // FullSimplify