Implicit Differentiation

We can find  
\[\frac{dy}{dx}\]
  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  
\[2xy+x^2 \frac{dy}{dx}-6y \frac{dy}{dx}=cos x \rightarrow \frac{dy}{dx}=\frac{cosx-2xy}{x^2-6y}\]

You have no rights to post comments