C38 · Unit C9 Parametric, polar and vector calculus
Parametric derivatives and second derivatives
BC extensionReview editionNot yet human-reviewed
- Time
- About 45 minutes, untimed
- Course framework reference
- AP 9.1–9.2 · BC
- Part of
Progress on this lesson counts in every pathway listed.
Understand the idea
For x equals x(t),y equals y(t), dy over dx equals (dy over dt)/(dx over dt) where dx over dt is not equal to 0. The second derivative is [d over dt(dy over dx)]/(dx over dt), not y seconds (t)/x seconds (t). Horizontal tangents require y minutes (t) equals 0 with x minutes (t) is not equal to 0; vertical tangents require x minutes (t) equals 0 with y minutes (t) is not equal to 0. If both vanish, inspect the local curve separately.
Worked example
For x equals t squared,y equals t cubed and t is not equal to 0, dy over dx equals 3t over 2. Then d squared y over dx squared equals (3 over 2)/(2t) equals 3/(4t). At t equals 2 the point is (4,8), slope 3 and second derivative 3 over 8.
Explore the mathematics
Parametric derivatives and second derivatives
Link two coordinate rates through their common parameter.
Scroll sideways to see the full diagram.
Predict before continuing
Both coordinates depend on the same parameter. How do you find dy/dx?
Read the complete explanation
- Both coordinates depend on the same parameter. How do you find dy/dx?
- Divide dy/dt=3t² by dx/dt=2t where the denominator is nonzero.
- dy/dx=3t/2. Differentiate with respect to t, then divide by dx/dt again for the second derivative.
- At t=4, slope=6; second derivative=3/(4t)=0.1875.
For the t≥0 branch x=t²,y=t³, the point t=2 is (4,8). Its tangent slope is 3 and its second derivative is 3/8. Parameter values t<0 trace a different branch below the x-axis.
Discuss: explain what each label represents, then draw a related example with different values.
A common misconception
The parameter derivative measures change per t; the graph derivative measures change per x. Dividing by x minutes converts between them.
Investigate and explain
Eliminate t in x equals t squared +1,y equals 2t and differentiate implicitly to verify the parametric result.
Identify one step in the worked example, name the mathematical fact that permits it, and state its domain conditions.