For an assignment, I've been asked to find the $\frac{dy}{dx}$ for the formula $2 \cos(4x) \sin(9y)=7$
The major problem I'm having with understanding implicit differentiation is understanding what distinguishes a typical $\frac{d}{dx}$ from $\frac{dy}{dx}$ An example I came across gave the process of $\frac{d}{dx}[xy^{2}]= x\frac{d}{dx}[y^{2}]+y^{2}\frac{d}{dx}[x]= x(2y\frac{dy}{dx}+y^{2}(1)= 2xy\frac{dy}{dx}+y^{2}$
Applying that approach to the above formula, I came up with
$2 \cos(4x) \sin (9y)=7 \rightarrow \frac{d}{dx}[2 \cos(4x) \sin(9y)]= \frac{d}{dx}(7)$
Applying the product rule to the elements:
$2[\cos(4x) \frac{d}{dx}\sin(9y)]+[\sin(9y)\frac{d}{dx} \cos(4x)]=0$ $2[\cos(4x)9 \cos(9y)\frac{dy}{dx}]+[\sin(9y)(-4 \sin(4x))\frac{dy}{dx}]=0$ $18 \cos(4x)\cos(9y)\frac{dy}{dx} + \sin(9y)-4 \sin(4x)\frac{dy}{dx}=0$
First, am I on the right track as far as assigning the correct $\frac{dy}{dx}$ terms? Assuming I am, how do I separate the $\frac{dy}{dx}$ terms from their respective functions? Because if I divide $18 \cos(4x)$ to isolate $\sin(9y)$, dividing against zero will produce zero.