Differentiation — FAQ

Use implicit differentiation when it is difficult or impossible to rearrange the equation to get $y$ on its own. For example, in $x^2 + xy + y^3 = 5$, solving for $y$ is very hard. It's much easier to differentiate term by term.

$\frac{dy}{dx}$ represents the rate of change of $y$ with respect to $x$; it's the gradient of the tangent to the curve on a standard $y$ vs $x$ graph. $\frac{dy}{dt}$ represents the rate of change of the variable $y$ with respect to time, $t$. It tells you how fast $y$ is changing over time, which is common in related rates problems.

Yes, you can. You can rewrite $\frac{u}{v}$ as $u \times v^{-1}$ and then apply the product rule along with the chain rule. However, the quotient rule is often more direct and less prone to error if you know the formula well. It's best to be proficient with both.

You can, but it is often much more complicated. Eliminating the parameter $t$ can lead to a very complex implicit equation. Using $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$ is usually a far more efficient method and is the expected technique in exams.