Question

If `x = yz` and `y = 2 sin(y + z)`, find `(dx)/(dy)`?

Answer 1

`dx/dy = arcsin(1/2y) + y/(2sqrt(1 - (1/2y)^2)) - 2y`

Explanation:

From our first equation, we know that `z = x/y`.

Therefore,

`y =2sin(y + x/y)`

Solving for `x` will prove quite easy (since we are differentiating with respect to `x`, this will probably make some sense).

`1/2y = sin(y + x/y)`

`arcsin(1/2y) = y + x/y`

`arcsin(1/2y) - y = x/y`

`yarcsin(1/2y) - y^2 = x`

Now we differentiate this and we will get our answer.

For the first term, `yarcsin(1/2y)`, we use the chain and product rules. We know the derivative of `arcsiny` to be `1/sqrt(1 - y^2)`, thus the derivative of `arcsin(1/2y) = 1/(2sqrt(1 - (1/2y)^2))`. By the product rule,

`arcsin(1/2y) + y/(2sqrt(1 -(1/2y)^2)) - 2y = (dx)/(dy)`

Hopefully this helps!