Implicit Differentiation

Tags

math

Is a form of differentiation targeting implicit functions.

For example consider the equation \( y^2 - 2x + \sin 3x = 16 \). Each of these can be differentiated independently, with any \( y \) terms also being multiplied by \( \frac{\,dy}{\,dx} \).

\begin{align*} \frac{\,d}{\,dx} (y^2) &= 2y \frac{\,dy}{\,dx} \
\frac{\,d}{\,dx} (-2x) &= -2 \
\frac{\,d}{\,dx} (\sin 3x) &= 3 \cos 3x \
\frac{\,d}{\,dx} (16) &= 0 \end{align*}

From this we can re-arrange to find \( \frac{\,dy}{\,dx} \)

\begin{align*} 2y \frac{\,dy}{\,dx} - 2 + 3\cos 3x &= 0 \
\frac{\,dy}{\,dx} = \frac{2 - 3 \cos 3x}{2y} \end{align*}