Implicit Differentiation combine 2 dy/dxs?

Question

When simplifying the expression: 4(x^2)(dy/dx)+14xy(dy/dx)=-7(y^2)-8xy

It says that dy/dx = (-7(y^2)-8xy)/4(x^2)+14xy

Why doesn't it equal (-7(y^2)-8xy)/(2(4(x^2)+14xy), because doesn't the two dy/dx's from the left side combine to make a 2(dy/dx)?

Answer 1

No, they combine to make $(4x^2+14xy)\frac{dy}{dx}$. If we had just had $\frac{dy}{dx}+\frac{dy}{dx}$, then that's really $1\cdot\frac{dy}{dx}+1\cdot\frac{dy}{dx}$ so they would combine to make $(1+1)\frac{dy}{dx}$, which is what you had in mind.

This reflects how we think about numbers in everyday speech as well. If you have three apples and five apples, then that makes eight apples (because three plus five is eight). The apples themselves do not later combine into two apples next to this.