Derivatives of parametric functions

Japanese school year: Math III

What you learn

This topic covers finding the derivative dy/dx of curves defined parametrically using the chain rule. It is essential for determining slopes, tangent vectors, and trajectories of geometric curves like cycloids and ellipses in kinematics. Prerequisites include a firm understanding of parametric representations and single-variable differentiation techniques.

Go to practice

Key points

This formula gives the slope $\frac{dy}{dx}$ when coordinates $x$ and $y$ are both defined by a parameter $t$ . Divide the derivative $\frac{dy}{dt}$ by $\frac{dx}{dt}$ .

\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} \quad \left(\frac{dx}{dt} \neq 0\right)

This formula calculates the second derivative $\frac{d^2y}{dx^2}$ for parametrically defined functions. Differentiate the first derivative $\frac{dy}{dx}$ with respect to $t$ , then divide again by $\frac{dx}{dt}$ .

\frac{d^2y}{dx^2} = \frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{\frac{dx}{dt}}

Choose a set to practice.

Practice 1Coming soon