This paper introduces a Caputo-type fractional derivative with respect to another function, studying its properties and applications. The derivative is defined using a general framework where the kernel is a function ψ(x), and the derivative operator is modified accordingly. The Caputo fractional derivative is defined as a fractional integral of the nth-order derivative of the function, where n is the smallest integer greater than α. The paper establishes a relationship between this operator and the Riemann–Liouville fractional derivative with respect to another function. It also proves several fundamental properties, including the semigroup law, a relationship between fractional integration and differentiation, and an integration by parts formula. The paper presents a numerical method for approximating the fractional derivative using a sum involving only integer-order derivatives. This method is shown to be efficient and applicable in various contexts. The paper also applies the fractional derivative to a population growth model, demonstrating that using different kernels for the fractional operator can provide a more accurate description of the dynamics of the model. The paper is organized into sections that define the Caputo-type fractional derivative, study its properties, compute its action on specific functions, and apply it to a population growth model. The main contributions of the paper include the generalization of the Caputo fractional derivative with respect to another function, the study of its properties, and the application of the derivative to a population growth model. The results show that the choice of kernel significantly affects the accuracy of the model, highlighting the importance of considering general forms of fractional operators in modeling real-world phenomena.This paper introduces a Caputo-type fractional derivative with respect to another function, studying its properties and applications. The derivative is defined using a general framework where the kernel is a function ψ(x), and the derivative operator is modified accordingly. The Caputo fractional derivative is defined as a fractional integral of the nth-order derivative of the function, where n is the smallest integer greater than α. The paper establishes a relationship between this operator and the Riemann–Liouville fractional derivative with respect to another function. It also proves several fundamental properties, including the semigroup law, a relationship between fractional integration and differentiation, and an integration by parts formula. The paper presents a numerical method for approximating the fractional derivative using a sum involving only integer-order derivatives. This method is shown to be efficient and applicable in various contexts. The paper also applies the fractional derivative to a population growth model, demonstrating that using different kernels for the fractional operator can provide a more accurate description of the dynamics of the model. The paper is organized into sections that define the Caputo-type fractional derivative, study its properties, compute its action on specific functions, and apply it to a population growth model. The main contributions of the paper include the generalization of the Caputo fractional derivative with respect to another function, the study of its properties, and the application of the derivative to a population growth model. The results show that the choice of kernel significantly affects the accuracy of the model, highlighting the importance of considering general forms of fractional operators in modeling real-world phenomena.