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

Parametric function is a function which $x$ and $y$ variables are seperated by certain parameter.

Example:

$x=10t$

$y=8t-3t^2$

$$\begin{array}{rl}\frac{dy}{dx}& =\frac{{\displaystyle \frac{dy}{dt}}}{{\displaystyle \frac{dx}{dt}}}\\ \\ \frac{dy}{dx}& =\frac{10}{8-6t}\end{array}$$

Implicit function is a function where $x$ and $y$ variables are on the same side, and they cannot be separated.

Examples

$x^3+y^5+x^2{y}^3-3x+4y+5=0$

Implicit function can be derived by differentiate the whole equation.

$$\begin{array}{rl}& x^3+y^5+x^2{y}^3-3x+4y+5=0\\ \\ & 3x^2+5y^4.y^{\prime }+2x.y^3+x^2.3y^2.y^{\prime }-3+4y^{\prime }+0=0\\ \\ & y^{\prime }(5y^4+3x^2{y}^2+4)=3-3x^2-2x{y}^3\\ \\ & y^{\prime }=\frac{3-3x^2-2x{y}^3}{5y^4+3x^2{y}^2+4}\end{array}$$

Notes:

• Derivative of $y$ is $y^{\prime }$, then derivative of $y^5$ is $5y^4.y^{\prime }$
• Derivative of $x^2.y^3$ can use product rule $u.v$