<p>In this article, the 1st-level left general fractional derivatives (1st-level LGFDs) and the 1st-level right general fractional derivatives (1st-level RGFDs) on a finite interval with singular kernels are defined and investigated for arbitrary order <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\alpha \in (n-1,n),\, n\in \mathbb {N}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="0.166667em" /> <mi>n</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation>. In fact, we establish and prove the first and the second fundamental theorems of fractional calculus (FC) for these derivatives with the associated integral operator. Furthermore, we introduce and prove representations for 1st-level LGFDs and 1st-level RGFDs in terms of LGFDs and RGFDs, respectively. For <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\alpha \in (0,1)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, we used these results to derive an integration by parts formula combining 1st-level LGFDs and 1st-level RGFDs. This integration by parts formula is crucial for studying the properties of the eigenvalues and eigenfunctions of boundary value problems involving left and right 1st-level GFDs.</p>