Parametric derivative formulas

College/University

Definition

Formulas for the first derivative and second derivitive often used to find the equation of a tangent line to a parametric curve. The derivative of Y (a dependant variable) with respect to X (a independant variable) is calculated when both X and Y depend on T (a third independant variable).

Worked examples

\(x = t^2, \; y = t^3 \quad \Rightarrow \quad \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{3t^2}{2t} = \frac{3t}{2}\)

Divide the derivative of y with respect to t by the derivative of x with respect to t.

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

The second derivative uses the chain rule: differentiate the first derivative with respect to t, then divide by dx/dt.

\(x = 2\cos t, \; y = 3\sin t \quad \Rightarrow \quad \frac{dy}{dx} = \frac{3\cos t}{-2\sin t} = -\frac{3}{2}\cot t\)

Use the parametric formula to find slope at any point on the ellipse.

Common mistakes

\(\frac{dy}{dx} = \frac{dx}{dt} \cdot \frac{dy}{dt}\) → \(\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}\) Divide dy/dt by dx/dt; don't multiply them.
\(\frac{d^2y}{dx^2} = \frac{\frac{d^2y}{dt^2}}{\frac{d^2x}{dt^2}}\) → \(\frac{d^2y}{dx^2} = \frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{\frac{dx}{dt}}\) The second derivative formula applies the parametric rule to dy/dx, not to the second derivatives of x and y separately.
\(\frac{dy}{dx}\) is undefined when \(\frac{dy}{dt} = 0\) → \(\frac{dy}{dx}\) is undefined when \(\frac{dx}{dt} = 0\) Division by zero occurs when the denominator dx/dt equals zero, not when the numerator is zero.

Where you'll use it next

Parametric derivatives are essential for finding tangent lines and analyzing motion along parametric curves in Calculus II, and for studying velocity and acceleration vectors in physics and multivariable calculus.

Found in 1 StudyPug lesson

Tangents and Concavity of Parametric Equations Explained

Calculus 2

Dive into the world of parametric equations, mastering tangents and concavity. Enhance your calculus skills with practical examples and real-world applications. Perfect for advanced students and enthusiasts.

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