Derivatives of Parametric Equations

What are Parametric Equations?

In many cases, instead of expressing y directly as a function of x, we describe both x and $y$ in terms of a third variable — called a parameter, often t (for time).

$$x=f(t) , y=g(t)$$

Find $$\frac{dy}{dx}$$​

We can’t directly differentiate y with respect to x since both depend on $t$.

$$frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$$

Example

$$x = t^{2}+1 , y=t^{3}+t$$

Step 1: Differentiate each with respect to t:

$$\frac{dx}{dt} = 2t , \frac{dy}{dx} = 3t^2+1$$

Step 2: Find $$\frac{dy}{dx}$$:

$$frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{3t^2+1}{2t}$$​​​​​

Real-Life Meaning

Parametric derivatives are used when:

• Describing motion (position given by x(t),y(t).

• Tracing curves where y is not a single-valued function of x.

• Modeling physics problems such as projectile motion.