# Parametric Derivative

How to solve the parametric derivative problems: formula, proof, example, and its solution.

## Formula

If *x* = *f*(*t*) and *y* = *g*(*t*),

then *dy*/*dx* = *g*'(*t*) / *f*'(*t*).*t* is called the parameter.

It connects the relationship between *x* and *y*.

## Proof

*x* = *f*(*t*)

So *dx*/*dt* = *f*'(*t*).

*y* = *g*(*t*)

So *dy*/*dt* = *g*'(*t*).

*dy*/*dx* = [*dy*/*dt*] / [*dx*/*dt*]

(Divide the numerator and the denominator by *dt*.)

Then [*dy*/*dt*] / [*dx*/*dt*] = *g*'(*t*) / *f*'(*t*).

## Example

*x* = *t*^{3} - 2*t*

So *dx*/*dt* = 3*t*^{2} - 2.

Derivatives of polynomials

*y* = *t*^{2} + 1

So *dx*/*dt* = 2*t*

So *dy*/*dx* = 2*t* / (3*t*^{2} - 2).

Put *t* = 1 into *dy*/*dx*.

Then [*dy*/*dx*]_{t = 1} = 2.