Differentiation

\(\dfrac {dy}{dx} = \dfrac {\dfrac {dy}{dt}}{\dfrac {dx}{dt}}\)

Parametric function is a function which \(x\) and \(y\) variables are seperated by certain parameter.

Example:

\(x = 10t\)

\(y = 8t - 3t^2\)

\begin{equation*} \begin{split} \frac {dy}{dx} & = \frac {\dfrac {dy}{dt}}{\dfrac {dx}{dt}}  \\\\ \frac {dy}{dx} & = \bbox[5px, border: 2px solid magenta] {\frac {10}{8 - 6t}} \end{split} \end{equation*}

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{equation*} \begin{split} & x^3 + y^5 + x^2 \: y^3 - 3x + 4y + 5 = 0 \\\\ & 3x^2 + 5y^4 \:.\: y' + 2x \:.\: y^3 + x^2 \:.\: 3y^2 \:.\: y' - 3 + 4y' + 0 = 0  \\\\ & y' (5y^4 + 3x^2 y^2 + 4) = 3 - 3x^2 - 2xy^3 \\\\ & y' = \bbox[5px, border: 2px solid magenta] {\frac {3 - 3x^2 - 2xy^3}{5y^4 + 3x^2 y^2 + 4}} \end{split} \end{equation*}

Notes:

• Derivative of \(y\) is \(y'\), then derivative of \(y^5\) is \(5y^4 \:.\: y'\)
• Derivative of \(x^2 \:.\: y^3\) can use product rule \(u \:.\: v\)