Derivatives of Parametric Functions

AP Calculus · Axiom Academy

LESSON Derivatives of Parametric Functions How to find dy/dx when x and y are both functions of t Step 1: The Problem We're Solving We have parametric equations and we want to find the slope of the tangent line: . The key insight: we can't just take because we have two different parameters! But we CAN use the chain rule: By dividing both sides by dt (and taking limits), we get our parametric derivative formula! Step 2: The Parametric Derivative Formula This is the most important formula for parametric calculus! It tells us: 1. Find dy/dt: the rate of change of y with respect to time 2. Find dx/dt: the rate of change of x with respect to time 3. Divide: dy/dx = (dy/dt) ÷ (dx/dt) Let's find dy/dx for the parametric curve: At t = 1, the slope of the tangent line is 2. Notice we got a single number for the slope, even though both x and y are functions of t! What if we want the second derivative? We can take the derivative of dy/dx: In other words, we treat dy/dx as another function of t, and differentiate it using the quotient rule! You're now ready to work with parametric derivatives in any problem! Practice finding slopes, second derivatives, and tangent lines to parametric curves.

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