Parametric Derivative

Parametric Derivative

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

Formula

If x = f(t), y = g(t), then dy/dx = g'(t)/f'(t).

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

Parametric Derivative: Proof of the Formula

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

For the given function, find dy/dx at t = 1. x = t^3 - 2t, y = t^2 + 1

x = t3 - 2t

So dx/dt = 3t2 - 2.

Derivatives of polynomials

y = t2 + 1

So dx/dt = 2t

So dy/dx = 2t / (3t2 - 2).

Put t = 1 into dy/dx.

Then [dy/dx]t = 1 = 2.