Hi could someone please explain to me how i would use the chain rule

dy/dx = dy/du(du/dx) to solve for
y= sin(x+y) - im looking for dy/dx
i don't understand the steps
thanks

let u = x + y

du/dx = 1 + dy/dx

then y = sin u
and dy/dx = cosu du/dx
= cosu (1 + dy/dx)
= cosu + dy/dxcosu
dy/dx - dy/dxcoxu = cosu
dy/dx = cosu/(1 - cosu)
= cos(x+y)/( 1 - cos(x+y)

personally I would bypass all that subbing and my second line would be

dy/dx = cos(x+y)(1 + dy/dx)
etc.

alright, but i don't know how i would now isolate for dy/dx?

Of course! The chain rule is used to differentiate composite functions, where one function is nested inside another. In this case, you have a composite function y = sin(x+y), and you are trying to find dy/dx.

To apply the chain rule, you need to break down the function into two parts: an outer function and an inner function. Let's define u = x + y. Therefore, we can rewrite the function as y = sin(u), where u = x + y.

Now, we can differentiate each part separately. The derivative of y with respect to u is dy/du, which is simply the derivative of sin(u). The derivative of sin(u) is cos(u).

Next, we need to find the derivative of u with respect to x, which is du/dx. Since u = x + y, we can differentiate both sides of the equation with respect to x. The derivative of x with respect to x is 1, and the derivative of y with respect to x is dy/dx. So we have du/dx = 1 + dy/dx.

Now we can use the chain rule formula:
dy/dx = (dy/du) * (du/dx)

Substituting the derivatives we found:
dy/dx = cos(u) * (1 + dy/dx)

Now, we can solve for dy/dx. First, we can multiply both sides by (1 + dy/dx):
dy/dx + dy/dx * (dy/dx) = cos(u) * (1 + dy/dx)

Simplifying:
dy/dx + (dy/dx)^2 = cos(u) + cos(u) * dy/dx

Rearranging the equation:
(dy/dx)^2 + dy/dx - cos(u) - cos(u)*dy/dx = 0

This is now a quadratic equation in terms of dy/dx. We can solve this equation to find dy/dx.