(PE3) Derivatives and Parametric Curves#
In this lesson, we are going to see how to calculate the:
first derivatives \(\dfrac{dy}{dx}\)
for a curve with differentiable parameteric equations.
Review Videos#
Derivatives#
First Derivative
Given differentiable parametric equations for \(x\) and \(y\), the first derivative can be calculated by:
\[
\dfrac{dy}{dx} = \dfrac{\quad \dfrac{dy}{dt}\quad }{\dfrac{dx}{dt}} \qquad \text{provided} \quad \dfrac{dx}{dt}\neq 0
\]
Second Derivative
Given differentiable parametric equations for \(x\) and \(y\), the second derivative can be calculated by:
\[
\dfrac{d^2y}{dx^2} = \dfrac{\dfrac{d}{dt}\left(\dfrac{dy}{dx}\right)}{\dfrac{dx}{dt}} \qquad \text{provided} \quad \dfrac{dx}{dt}\neq 0
\]
Example 1#
Calculate the first derivative \(\dfrac{dy}{dx}\) given parametric equations:
\[
x=t^2+t\qquad \qquad y=t^3-3t
\]
Example 2#
Find the equation of the tangent line to the curve at the point \((6,2)\).
\[
x=t^2+t\qquad \qquad y=t^3-3t
\]
