For a parametric curve, dy/dx is (dy/dt) ÷ (dx/dt). Differentiate each coordinate against t and divide.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Why it works
The Chain Rule gives dy/dt = (dy/dx)(dx/dt). Rearranging isolates dy/dx without ever eliminating t.
Where the tangent is vertical
A vertical tangent occurs where dx/dt is zero and dy/dt is not. A horizontal one is the reverse.
The second derivative
d²y/dx² is the derivative of dy/dx with respect to t, divided again by dx/dt. Dividing only once is the standard error.
Arc length
The length is the integral of √((dx/dt)² + (dy/dt)²) dt. It is Pythagoras applied to an infinitesimal step.
Speed
That same square root is the speed of the particle. Integrating speed over time gives distance travelled.
Slope without eliminating t
dy/dx = (dy/dt)/(dx/dt), provided dx/dt is not zero. The chain rule supplies this directly, so the tangent slope is available without converting to Cartesian form.
The second derivative is not the ratio of seconds
d²y/dx² is the derivative of dy/dx with respect to t, divided by dx/dt. It is emphatically not (d²y/dt²)/(d²x/dt²), and that mistaken shortcut is the standard error in the topic.
Speed is the magnitude of velocity
Speed is √((dx/dt)² + (dy/dt)²). Velocity is the vector of the two rates; speed is its length and is never negative. The exam distinguishes the two deliberately.
Arc length integrates speed
Distance travelled is the integral of speed with respect to t. That is why the arc length formula has a square root of squared derivatives — it is summing speed over time.
Step 2: Try It Yourself
Tap and try it out.
- Point(2, 4)
- Slope of the tangent4
Step 3: Watch an Example
One step at a time.
Watch Rosa Find a Parametric Slope
Rosa has x = t² and y = t³, and needs dy/dx at t = 2.
- Step 1
Differentiating gives dx/dt = 2t and dy/dt = 3t².
Step 4: Your Turn
Practice makes it stick.
The Slope
Problem 1 of 2
dy/dt = 12 and dx/dt = 4. What is dy/dx?
The Speed
Problem 2 of 2
dx/dt = 3 and dy/dt = 4. What is the speed?
Differentiate Parametrically
1 of 8
dy/dt = 10, dx/dt = 5. What is dy/dx?
2 of 8
dx/dt = 6, dy/dt = 8. What is the speed?
3 of 8
x = t², so dx/dt = 2t. What is dx/dt at t = 5?
4 of 8
y = t³, so dy/dt = 3t². What is dy/dt at t = 2?
5 of 8
dx/dt = 0 and dy/dt = 5. Is the tangent vertical or horizontal? 1 vertical, 2 horizontal.
6 of 8
dx/dt = 5 and dy/dt = 0. Vertical or horizontal? 1 or 2?
7 of 8
Put the process for the second derivative in order.
- 1Differentiate that expression with respect to t.
- 2Divide the result by dx/dt again.
- 3Simplify to get the second derivative.
- 4Compute dy/dx as the ratio of the two rates.
8 of 8
dx/dt = 5 and dy/dt = 12. What is the speed?
Step 5: Quick Check
Show what you know.
Question 1 of 2
dy/dt = 18 and dx/dt = 6. What is dy/dx?
Question 2 of 2
What is the common error when finding the second derivative?
What You Learned
- dy/dx is (dy/dt) ÷ (dx/dt).
- The second derivative divides by dx/dt a second time.
- Speed and arc length both use √((dx/dt)² + (dy/dt)²).