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:

$x=\sin t$

$y=t$

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

$x^{\prime }=\cos t$

$y^{\prime }=1$

In general, we say that if

$x=f(t)$ and $y=g(t)$ then:

$x^{\prime }=f^{\prime }(t)$ and $y^{\prime }=g^{\prime }(t)$

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 change, 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:

$\frac{y^{\prime }}{x^{\prime }}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{dy}{dx}$

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' and 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 that it may seem at first glance. When you have take the derivative of $\frac{dy}{dx}$ in terms of t, you are left with $\frac{\frac{d^{2}y}{dx}}{dt}$:

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

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

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

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

$\frac{d}{dt}[\frac{dy}{dx}]*\frac{dt}{dx}$

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

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