Second Derivative (Parametric)
Unit 9 · Parametric, Polar & Vector
Second Derivative of Parametric Curves
The second derivative of a parametric curve is d²y/dx² = [d(dy/dx)/dt] / (dx/dt). First compute dy/dx as a function of t, differentiate it with respect to t, then divide by dx/dt. The result determines concavity.
Second Derivative Formula
Concave up if d²y/dx² > 0; concave down if d²y/dx² < 0
Caution: Do NOT compute (d²y/dt²)/(d²x/dt²) — that is incorrect. Differentiate dy/dx (as a function of t) with respect to t, then divide by dx/dt.
Type 1
Computing the Second Derivative
Find dy/dx, differentiate it with respect to t, then divide by dx/dt.
Example 1
For x = t², y = t³, find d²y/dx².
Example 2
For x = t³, y = t² − 1, find d²y/dx² and evaluate at t = 1.
Practice more of this type— AI-generated · infinite problems
Generate Problems →Type 2
Concavity of a Parametric Curve
Evaluate d²y/dx² at a specific t value to determine whether the curve is concave up or concave down at that point.
Example 3
For x = eᵗ, y = t², determine concavity at t = 0.
Example 4
For x = eᵗ, y = e^{2t}, find d²y/dx².
Practice more of this type— AI-generated · infinite problems
Generate Problems →