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:

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

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

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

In general, we say that if

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

then:

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

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 ${\frac {\frac {d^{2}y}{dx}}{dt}}$ :

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

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

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

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 $t$ :

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