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Finding a simple root for a nonlinear equation <inline-formula> <math display="inline"> <semantics> <mrow> <mi>f</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> <mo>=</mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>f</mi> <mo>:</mo> <mi>I</mi> <mo>&sube;</mo> <mi mathvariant="double-struck">R</mi> <mo>&#8594;</mo> <mi mathvariant="double-struck">R</mi> </mrow> </semantics> </math> </inline-formula> has always been of much interest due to its wide applications in many fields of science and engineering. Newton&#8217;s method is usually applied to solve this kind of problems. In this paper, for such problems, we present a family of optimal derivative-free root finding methods of arbitrary high order based on inverse interpolation and modify it by using a transformation of first order derivative. Convergence analysis of the modified methods confirms that the optimal order of convergence is preserved according to the Kung-Traub conjecture. To examine the effectiveness and significance of the newly developed methods numerically, several nonlinear equations including the van der Waals equation are tested.