s this incorrect and why? $$ \frac{d}{dx}[tan^{-1}(x^2) = ln(y) + e^{\cos{x}}]$$
This is a question in a practice exam for using implicit differentiation. Given that the relation is not explicitly solved for $y$, it seems convenient to use this notation to mean "take the derivative of what is in the brackets". But, is it correct? I think if it is incorrect, it is an abuse of notation at best, but given the differentiation is a linear operator, I think it may be correct.
It is poor notation in my opinion - "differentiating" the equals sign doesn't make sense. It would be clearer and more correct to write: $$\tan^{-1}\left(x^2\right)=\ln y+e^{\cos(x)}\implies\frac{d}{dx}\tan^{-1}\left(x^2\right)=\frac{d}{dx}\left[\ln y+e^{\cos(x)}\right],$$ which emphasizes that you are applying the differential operator to both sides and actually makes sense as an equality.