\[\frac{dy}{dx}\]
even when \[y\]
is not given as a function of \[x\]
. This is called implicit differentiation.Example:
\[x^2y-3y^2=sinx\]
.We can differentiate the first term
\[x^2y\]
using The Product Rule and The Chain Rule.\[\frac{d(x^2y)}{dx}=\frac{d(x^2)}{dx}y+x^2 \frac{dy}{dx}=2xy+x^2 \frac{dy}{dx}\]
We can differentiate the second term
\[3y^2\]
using the Chain Rule\[\frac{d(3y^2)}{dx}=6y \frac{dy}{dx}\]
We have
\[2xy+x^2 \frac{dy}{dx}-6y \frac{dy}{dx}=cos x \rightarrow \frac{dy}{dx}=\frac{cosx-2xy}{x^2-6y}\]