Mathematics and Statistics · Ch 3 — Differentiation

Derivatives of Parametric Functions

Sometimes both $x$ and $y$ are given separately in terms of a third variable $t$ (the parameter), as in $x = at^2,\ y = 2at$ . Such a pair is a parametric representation of a curve. To find $\dfrac{dy}{dx}$ without eliminating $t$ , use:

\frac{dy}{dx} = \frac{\;\dfrac{dy}{dt}\;}{\;\dfrac{dx}{dt}\;} \qquad \left(\text{provided } \frac{dx}{dt} \neq 0\right).

That is, differentiate $y$ and $x$ separately with respect to the parameter $t$ , then divide. This follows from the chain rule: $\dfrac{dy}{dx}=\dfrac{dy}{dt}\cdot\dfrac{dt}{dx}=\dfrac{dy}{dt}\big/\dfrac{dx}{dt}$ .

Method — parametric differentiation:

Differentiate $y$ with respect to $t$ to get $\dfrac{dy}{dt}$ .
Differentiate $x$ with respect to $t$ to get $\dfrac{dx}{dt}$ .
Divide: $\dfrac{dy}{dx} = \dfrac{dy/dt}{dx/dt}$ .
Simplify; the answer is usually left in terms of the parameter $t$ (this is perfectly acceptable).

Illustration. For $x = t^2,\ y = t^3$ : $\dfrac{dx}{dt}=2t$ and $\dfrac{dy}{dt}=3t^2$ , so $\dfrac{dy}{dx}=\dfrac{3t^2}{2t}=\dfrac{3t}{2}$ .

Note

Divide $\dfrac{dy}{dt}$ by $\dfrac{dx}{dt}$ — Not the Other Way, and Never Divide the Functions Themselves …

Definition 10Parametric differentiation

When $x=f(t)$ and $y=g(t)$ are given in terms of a parameter $t$ , then $\dfrac{dy}{dx}=\dfrac{dy/dt}{dx/dt}$ (with $\tfrac{dx}{dt}\neq0$ ). Differentiate each with respect to $t$ and divide; the …

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