# Derivative of Parametric functions

### Parametric Equations:

• Parametric equations represent a relationship between two variables, often denoted as $x$ and $y$, where each variable is expressed in terms of a third variable, usually $t$, known as the parameter.
• Common parametric equations take the form: $x=f\left(t\right)$ $y=g\left(t\right)$

### Derivative of Parametric Functions:

• To find the derivative of a parametric curve given by $x=f\left(t\right)$ and $y=g\left(t\right)$, the derivative $\frac{dy}{dx}$ is calculated using the chain rule.
• The derivative $\frac{dy}{dx}$ represents the slope of the curve at any given point.

### Steps to Find $\frac{dy}{dx}$:

1. Derive both x and y with respect to $t$: $\frac{dx}{dt}={f}^{\mathrm{\prime }}\left(t\right)$ $\frac{dy}{dt}={g}^{\mathrm{\prime }}\left(t\right)$

2. Calculate $\frac{dy}{dx}$ using the chain rule: $\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{{g}^{\mathrm{\prime }}\left(t\right)}{{f}^{\mathrm{\prime }}\left(t\right)}$

### Example:

Given parametric equations $x=\mathrm{cos}\left(t\right)$ and $y=\mathrm{sin}\left(t\right)$:

1. Find $\frac{dy}{dx}$:

$\frac{dx}{dt}=-\mathrm{sin}\left(t\right)$
$\frac{dy}{dt}=\mathrm{cos}\left(t\right)$

2. Calculate $\frac{dy}{dx}$:

$\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{\mathrm{cos}\left(t\right)}{-\mathrm{sin}\left(t\right)}=-\mathrm{tan}\left(t\right)$

### Special Cases:

• Vertical Tangents:
• When $\frac{dx}{dt}=0$ but $\frac{dy}{dt}\mathrm{\ne }0$, the curve has a vertical tangent.
• Horizontal Tangents:
• When $\frac{dy}{dt}=0$ but $\frac{dx}{dt}\mathrm{\ne }0$, the curve has a horizontal tangent.

### Tips:

• Parametric Differentiation Rule:

• Sometimes it's convenient to express $y$ explicitly as a function of $x$ and differentiate using implicit differentiation.
• Substitute for $t$ or Express in terms of $x$ or $y$:

• Substituting $t$ with expressions involving $x$ or $y$ can help simplify calculations or analyze the curve's behavior at specific points.