Question

If `z^2 = x^2 + y^2, dx/dt = 2,` and `dy/dt = 3`, what is `dz/dt` when `x = 5` and `y = 12`?

Answer 1

`dz/dt=46/13`

Explanation:

We must differentiate implicitly with respect to `t`.

`d/(dt)[z^2=x^2+y^2]rarr2z(dz)/dt=2x(dx)/dt+2y(dy)/dt`

We know that `x=5` and `y=12`, so we can use the original equation to determine that `z=13`.

We can plug in all the values that we now know:
`2(13)dz/dt=2(5)2+2(12)3`
`dz/dt=46/13`

Note that `z` could also equal `-13`, but I am assuming that the physical restrictions behind this problem would not allow such an answer. If I am incorrect, `dz/dt=+-46/13`.