Parametric Derivatives
dy/dx from x(t), y(t).
Key idea
dy/dx from x(t), y(t).
In depth
For x = x(t), y = y(t), dy/dx = (dy/dt)/(dx/dt). The second derivative is d/dt(dy/dx) divided by dx/dt. Tangent lines are vertical when dx/dt = 0.
Worked example
x = t², y = t³: dy/dx = 3t²/2t = 3t/2. At t = 2, slope = 3.
Formula
Real-world uses
- Finding the slope of a moving object’s path, such as a drone or a planet orbit
Common mistake
Computing d²y/dx² as (d²y/dt²)/(d²x/dt²).
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Parametric Derivatives?
dy/dx from x(t), y(t). It is a topic at the expert level in Calculus.
What is the formula for Parametric Derivatives?
The key formula is dy/dx = (dy/dt)/(dx/dt).
Where is Parametric Derivatives used in real life?
Real-world examples: Finding the slope of a moving object’s path, such as a drone or a planet orbit.
What should I learn before Parametric Derivatives?
Make sure you are comfortable with Taylor’s Remainder, the previous topic in Calculus.
What is a common mistake with Parametric Derivatives?
Computing d²y/dx² as (d²y/dt²)/(d²x/dt²).
How can I practise Parametric Derivatives?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Parametric Derivatives?
Move on to Polar Calculus next.
Related expert topics in Calculus
- Derivatives of Inverse Trig – Derivatives of arcsin, arccos, arctan.
- Implicit Differentiation – Differentiate both sides of F(x,y)=0 treating y as a function of x.
- Logarithmic Differentiation – Take logs before differentiating products and powers.
- Related Rates – Link rates of change of connected quantities.
- Linear Approximation – Approximate f near a using its tangent line.
- Differentials – Use dy = f′(x)dx for small changes.