Parametric derivatives
Differentiate a curve whose coordinates depend on a parameter.
Parametric derivatives is one of 2 parametric formulas in the calculus section of this library, and it is used at ap · university level.
Why parametric derivatives works
Both coordinates move as the parameter runs, and dividing their two rates cancels the parameter, leaving rise per unit of run at that instant. The second derivative repeats the same trick on the first one, which is why it must again be divided by the horizontal rate.
What each symbol means
$x=x(t)$ and $y=y(t)$.
Parametric derivatives: when it holds
$dx/dt\ne0$ at the point for these quotients.
When it stops applying
Where the horizontal rate is 0 the quotient divides by zero and reports no slope. At the cusp of the curve given by x = t² and y = t³ the point momentarily stops moving sideways, so that instant has to be examined directly rather than through the formula.
Parametric derivatives: a worked example
If $x=t^2,y=t^3$, then $dy/dx=3t/2$ for $t\ne0$.
The mistake to avoid
What people do: Building the second derivative as the second vertical rate divided by the second horizontal rate.
Why it goes wrong: Copying the pattern of the first derivative is not what the chain rule allows, and the result is a different quantity with no geometric meaning.
Do this instead: Find the first derivative, differentiate that whole expression with respect to the parameter, then divide by the horizontal rate once more.
Parametric derivatives: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $x=x(t)$ and $y=y(t)$.
- Check the conditions before substituting. $dx/dt\ne0$ at the point for these quotients.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most calculus slips.
Where this formula fits
- Subject
- Calculus formulas — 42 entries in this library
- Topic
- Parametric
- Level
- AP · University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where parametric derivatives comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Derivative Rules — the lesson behind this formula: differentiate sums, products, quotients, and compositions.
- Calculus Calculator — check your substitution and the value it produces.
- Study calculus — the subject guide that explains the ideas these formulas compress.
- Calculus Practice — questions that make you retrieve the formula instead of recognising it.
- All 42 calculus formulas — the full grouped reference, or the complete formula library.
Questions about parametric derivatives
Why does the parameter cancel?
Because the slope asks how much the height changes per unit of horizontal travel, and both of those are being measured against the same parameter, which divides out.
How do I find horizontal and vertical tangents?
A horizontal tangent needs the vertical rate to be 0 while the horizontal rate is not, and a vertical tangent needs the opposite pairing.
Does the second derivative still describe concavity?
Yes, positive means the curve bends upward, but only over stretches where the horizontal rate keeps one sign so the point moves steadily in one direction.
What if the curve passes through a point twice?
Each visit happens at its own parameter value and may have its own slope, so ask for the derivative at a parameter value rather than at a location.