Implicit derivative

Template:Orphan Implicit derivatives are derivatives of implicit functions. This means that they are not in the form of $y=f(x)$ (explicit function), and are instead in the form $0=f(x,y)$ (implicit function). It might not be possible to rearrange the function into the form $y=f(x)$. To use implicit differentiation, we use the chain rule,

$\frac{\mathrm{d}t}{\mathrm{d}x}=\frac{\mathrm{d}t}{\mathrm{d}y}\frac{\mathrm{d}y}{\mathrm{d}x}$

If we let $t=f(y)$, then,

$\frac{\mathrm{d}}{\mathrm{d}x}f(y)=\frac{\mathrm{d}}{\mathrm{d}y}f(y)\frac{\mathrm{d}y}{\mathrm{d}x}=f^{\prime }(y)\frac{\mathrm{d}y}{\mathrm{d}x}$

Example

$x=6y^2+5x^4-y^3$

$1=6\frac{\mathrm{d}}{\mathrm{d}x}{y}^2+20x^3-\frac{\mathrm{d}}{\mathrm{d}x}{y}^3$

Which we can work out to be equivalent to, using the above,

$1-20x^3=6\cdot 2y\frac{\mathrm{d}y}{\mathrm{d}x}-3y^2\frac{\mathrm{d}y}{\mathrm{d}x}$

Then we can isolate $\frac{\mathrm{d}y}{\mathrm{d}x}$

$1-20x^3=\frac{\mathrm{d}y}{\mathrm{d}x}[12y-3y^2]$

Then divide to get,

$\frac{1-20x^3}{12y-3y^2}=\frac{\mathrm{d}y}{\mathrm{d}x}$