Exist Contravariant derivative? I'm confused about the index representation, If the index up represents a contravariant tensor I can do this with the derivative and get it contravariant derivative?Or when it is said that, it is covariant or contravariant by "essence"(before the contravariant and covariant formulation like the gradient Which is  transformed covariant).
I can't get an image because I don't have a score or whatever then I'll try to write,
(Contravariant)  $T^{i'}=\dfrac{\partial x^{i'}}{\partial x_j}T^j$
(covariant).       $V_{i'}=\dfrac{\partial x^j}{\partial x^{i'}}V_j$
Then $g^{ij}\partial_j=\partial^i$, I can transformed it contravariant, For it has index above, But the derivative is always covariant, im confused, what is $\partial ^j$.
 A: There is -as we know- a covariant derivative but not a contravariant one. A vector with upper/lower index is called contravariant/covariant, and -as we know- indices are raised/lowered by the metric: $V^\mu=g^{\mu\nu}V_\nu\,.$ This works for the partial derivative as well: $\partial^\mu=g^{\mu\nu}\partial_\nu\,$ and also for the covariant derivative $\nabla^\mu=g^{\mu\nu}\nabla_\nu\,.$  There is a clash in the nomenclature: $V^\mu$ is a called contravariant vector, but $\partial^\mu,\nabla^\mu$ are not called contravariant derivatives.
An acceptable name for $\nabla^\mu$ is covariant derivative with an upper index.
The covariant derivative is related to the partial derivative by
$$
\nabla_\mu V^\nu=\partial_\mu V^\nu+\Gamma^\nu_{\mu\lambda}V^\lambda
$$
(Carroll (2.19) on p. 45, (3.2) on p. 56) and we have the transformation laws
$$
V^{\nu'}=\frac{\partial x^{\nu'}}{\partial x^{\nu}} V^\nu\,,\quad\quad
\nabla_{\mu'}=\frac{\partial x^\mu}{\partial x^{\mu'}} \nabla_\mu\,,\quad\quad
\nabla_{\mu'}V^{\nu'}=\frac{\partial x^\mu}{\partial x^{\mu'}}\frac{\partial x^{\nu'}}{\partial x^{\nu}}\nabla_\mu V^\nu\,.
$$
Because the primed $\partial x^{\nu'}$ is in the numerator of the first law this is called a contravariant transformation. Likewise, the second law is a covariant transformation. It is very instructive to work this out explicitly by looking at a Lorentz transformation. The third law is a combination of the first and second law. It is neither covariant nor contravariant (rather mixed) but: because the derivatives operator $\nabla$ can be pulled through the transformation
$$
\frac{\partial x^\mu}{\partial x^{\mu'}}\frac{\partial x^{\nu'}}{\partial x^{\nu}}
$$
it acts as if that term was a constant. The plain partial derivative can't do that unless the Christoffel symbols $\Gamma$ are all zero (no curvature). This invariance of $\nabla$ is the reason why it is called (somehow unfortunately) covariant derivative. Some people prefer to call it invariant derivative instead. Other people today avoid to call $V_\mu$ covariant, resp. $V^\mu$ contravariant, or are just happy with the understanding that it tells where the index is sitting.
This link, or for example, H. Weyl's book Space Time Matter give some intuition and historical background about these nomenclatures.
