Differentiation — common mistakes

Memorise the formula for the parametric second derivative. A very common mistake is to calculate (d²y/dt²) / (d²x/dt²), which is incorrect. Always remember you are differentiating dy/dx (which is a function of $t$) with respect to $x$, so you must apply the chain rule, which introduces the 1/(dx/dt) factor.

It's called 'implicit' because the relationship between $y$ and $x$ is implied by the equation, rather than being explicitly stated as $y = f(x)$. We work with this implied relationship directly.

The parameter $t$ can represent many things. In physics, it often represents time, with the equations describing the motion of an object. In pure mathematics, it can be thought of as a variable that 'draws' the curve as it changes. For example, for the parametric equations of a circle $x = cos(t), y = sin(t)$, $t$ represents the angle.

The form of the equation tells you which method to use. If you have a single equation relating $x$ and $y$ (e.g., $x^{2} + xy + y^{3} = 2$), you must use implicit differentiation. If you are given $x$ and $y$ separately as functions of a third variable $t$ (e.g., $x = t^{2}$, $y = 2t$), you must use parametric differentiation.

Certainly. The key is to remember what d²y/dx² means: it's the derivative with respect to x of dy/dx. Since your expression for dy/dx is in terms of $t$, you can't differentiate it with respect to $x$ directly. You must use the chain rule: $d/dx(A) = (d/dt(A)) * (dt/dx)$. Here, $A$ is your dy/dx expression. This gives d/dt(dy/dx) * (dt/dx). Since $dt/dx = 1/(dx/dt)$, the full formula becomes [d/dt(dy/dx)] / (dx/dt).