concept of intrinsic derivative in light of covariant derivative
Andrew Mclaughlin 
What is the basic difference between ordinary differentiation , covariant differentiation and intrinsic differentiation ?
While taking a first course in tensor analysis , I was taught that ordinary derivative of a tensor may not produce a tensor necessarily . So we need to define a covariant derivative in order to fulfill that gap . But I am confused why we need to define an intrinsic one particularly . It is defined that for some mixed tensor $A^i_j$ having components as functions of a parameter $t$ we have the intrinsic derivative $$\frac{\delta A^i_j}{\delta t}=A^i_{j, \ k}\frac{dx^k}{dt}=\left(\frac{\partial A^i_j}{\partial x^k}+\Gamma^i_{lk}A^l_j-\Gamma^l_{jk}A^i_l\right)\frac{dx^k}{dt}$$ $$\implies\frac{\delta A^i_j}{\delta t}=\frac{dA^i_j}{dt}+(\Gamma^i_{lk}A^l_j-\Gamma^l_{jk}A^i_l)\frac{dx^k}{dt}$$ where $x^k(t)$ are coordinates in the manifold .
So what does it make the difference between a covariant derivative and intrinsic derivative despite the fact that both produce a tensor ? In fact , why do we even need to define it as presented above ? Any help is appreciated .
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