Calculus/Parametric Differentiation

Taking Derivatives of Parametric Systems
Just as we are able to differentiate functions of $$x$$, we are able to differentiate $$x$$ and $$y$$ , which are functions of $$t$$. Consider:

$$\begin{align}x&=\sin(t)\\y&=t\end{align}$$

We would find the derivative of $$x$$ with respect to $$t$$, and the derivative of $$y$$ with respect to $$t$$ :

$$\begin{align}x'&=\cos(t)\\y'&=1\end{align}$$

In general, we say that if

$$\begin{align}x&=x(t)\\y&=y(t)\end{align}$$

then:

$$\begin{align}x'&=x'(t)\\y'&=y'(t)\end{align}$$

It's that simple.

This process works for any amount of variables.

Slope of Parametric Equations
In the above process, $$x'$$ has told us only the rate at which $$x$$ is changing, not the rate for $$y$$, and vice versa. Neither is the slope.

In order to find the slope, we need something of the form $$\frac{dy}{dx}$$.

We can discover a way to do this by simple algebraic manipulation:

$$\frac{y'}{x'}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{dy}{dx}$$

So, for the example in section 1, the slope at any time $$t$$ :

$$\frac{1}{\cos(t)}=\sec(t)$$

In order to find a vertical tangent line, set the horizontal change, or $$x'$$, equal to 0 and solve.

In order to find a horizontal tangent line, set the vertical change, or $$y'$$, equal to 0 and solve.

If there is a time when both $$x',y'$$ are 0, that point is called a singular point.

Concavity of Parametric Equations
Solving for the second derivative of a parametric equation can be more complex than it may seem at first glance.

When you have take the derivative of $$\frac{dy}{dx}$$ in terms of $$t$$, you are left with $$\frac{\frac{d^2y}{dx}}{dt}$$ :

$$\frac{d}{dt}\left[\frac{dy}{dx}\right]=\frac{\frac{d^2y}{dx}}{dt}$$.

By multiplying this expression by $$\frac{dt}{dx}$$, we are able to solve for the second derivative of the parametric equation:

$$\frac{\frac{d^2y}{dx}}{dt}\times\frac{dt}{dx}=\frac{d^2y}{dx^2}$$.

Thus, the concavity of a parametric equation can be described as:

$$\frac{d}{dt}\left[\frac{dy}{dx}\right]\times\frac{dt}{dx}$$

So for the example in sections 1 and 2, the concavity at any time $$t$$ :

$$\frac{d}{dt}[\csc(t)]\times\cos(t)=-\csc^2(t)\times\cos(t)$$