How to find the derivative of a parametric curve

What is the derivative of a parametric curve?

Given a parametric curve where our function is defined by two equations, one for $x$ and one for $y$ , and both of them in terms of a parameter $t$ ,

$x=f(t)$

$y=g(t)$

we calculate the derivative of the parametric curve using the formula

$\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}$

where $dy/dx$ is the first derivative of the parametric curve, $dx/dt$ is the derivative of $x=f(t)$ and $dy/dt$ is the derivative of $y=g(t)$ .

Example

Find the derivative of the parametric curve.

$x=3t^4-6$

$y=2e^{4t}$

We’ll start by finding $dy/dt$ and $dx/dt$ .

$y=2e^{4t}$

$\frac{dy}{dt}=8e^{4t}$

and

$x=3t^4-6$

$\frac{dx}{dt}=12t^3$

Plugging these into the derivative formula for $dy/dx$ , we get

$\frac{dy}{dx}=\frac{8e^{4t}}{12t^3}$

$\frac{dy}{dx}=\frac{2e^{4t}}{3t^3}$