Find $\frac{dy}{dx}$ given that $\sin(8x+y)=18x$. Find $\frac{dy}{dx}$ given that $\sin(8x+y)=18x$.
So far, I've done the chain rule and find the derivatives for both sides so I got:
$\cos(8x+y)(8+\frac{dy}{dx})=18$
From here I'm lost, do I subtract 18 from both sides and then multiply it by 8 so I get:
$-144\cos(8x+y)(\frac{dy}{dx})=0$  ?
Please Help!!!
 A: Hint: $$y=\arcsin (18x) - 8x.$$
A: $$
\cos(8x+y)(8+\frac{dy}{dx})=18
$$
For the other side
$$
\cos(8x+y)=\sqrt{1-\sin^2(8x+y)}=\sqrt{1-(18x)^2}
$$
Then
$$
\frac{dy}{dx}=\frac{18}{\sqrt{1-(18x)^2}}-8
$$
A: Your problems are with algebraic manipulation.  You have a linear equation to solve for $\frac{dy}{dx}$.  If you were trying to solve $3(5+T)=7$ for $T$, you could divide by $3$ to get $5+T=\frac{7}{3}$, then subtract $5$ to get $T=\frac{7}{3}-5$ (which you might rewrite as $T=-\frac{8}{3}$).  Or, you could distribute the multiplication on the left first, to get $15+3T=7$, then subtract $15$ to get $3T=-8$, then divide by $3$ to get $T=-\frac{8}{3}$.
Your equation has the same general form, in that $\dfrac{dy}{dx}$ plays the role of $T$ above, and the other parts can be thought of as numbers to subtract, add, muliply, or divide according to correct algebraic rules.

From here I'm lost, do I subtract 18 from both sides and then multiply it by 8 so I get:
$-144\cos(8x+y)(\frac{dy}{dx})=0$  ?

Here you are saying that you got $\cos(8x+y)(8+\frac{dy}{dx})-18=-144\cos(8x+y)(\frac{dy}{dx})$, and that would be like saying $3(5+T)-7 = -35\cdot3(T)$; it does not follow from principles of arithmetic that numbers can be rearranged in such ways.  You can use principles like $(ab)/a = b$, $(a+b)-a = b$, and $a(b+c)= ab+ac$.
You want to isolate $\dfrac{dy}{dx}$; in order to do so, you can divide by $\cos(8x+y)$ to get $8+\frac{dy}{dx}=\dfrac{18}{\cos(8x+y)}$.  Then you can subtract $8$ to get $\frac{dy}{dx}=\dfrac{18}{\cos(8x+y)}-8$.  André Nicolas points out in a comment how you can solve entirely in terms of $x$, similar to Pocho la pantera's answer, except that the sign of $\cos(8x+y)$ is not clear, hence which root to take is not clear.
