Question

Question #da527

Answer 1

I think there is an `=` sign missing but....

Explanation:

Here you have an Implicit Derivation; in this case the function is kind of "hidden" so that you cannot really write it in the normal, explicit: `y=f(x)`.
Our idea is that `y` represents a function of `x` so that when you derive you must include the term `(dy)/(dx)` to indicate the fact the `y` is, in itsef, a function of `x`.
So for example if you have `y^2` this derived will give you:
`2y(dy)/(dx)`

In your case I suspect that probably there should be a `=` sign somewhere such as:
`8xy+16sin(y)=0`
or other (you need the `=` sign to indicate the functional dependence such as: "this depends from that");

but anyway, if you ONLY derive your expression you should get:

`8y+8x(dy)/(dx)+16cos(y)(dy)/(dx)`

Now it ends here...on the other hand if you consider the original as:
`8xy+16sin(y)=0`
you'll get:
`8y+8x(dy)/(dx)+16cos(y)(dy)/(dx)=0`
you can now collect `(dy)/(dx)`, rearrange and write:
`(dy)/(dx)=-(cancel(8)y)/(cancel(8)(x+2cos(y)))=-(y)/((x+2cos(y))`