This paper introduces and studies a Caputo-type fractional derivative with respect to another function. The authors define the left and right $\psi$-Caputo fractional derivatives and explore their properties, including semigroup laws, integration by parts, Fermat's Theorem, and Taylor's Theorem. They also present a numerical method to approximate these fractional derivatives using sums involving integer-order derivatives. The efficiency and applicability of this method are demonstrated through examples. Additionally, the paper applies the fractional derivative to a population growth model, showing that using different kernels for the fractional operator can lead to more accurate models. The main results include the boundedness of the $\psi$-Caputo fractional derivatives, a relationship between the fractional derivative and integral, and various theorems that establish the properties of this operator.This paper introduces and studies a Caputo-type fractional derivative with respect to another function. The authors define the left and right $\psi$-Caputo fractional derivatives and explore their properties, including semigroup laws, integration by parts, Fermat's Theorem, and Taylor's Theorem. They also present a numerical method to approximate these fractional derivatives using sums involving integer-order derivatives. The efficiency and applicability of this method are demonstrated through examples. Additionally, the paper applies the fractional derivative to a population growth model, showing that using different kernels for the fractional operator can lead to more accurate models. The main results include the boundedness of the $\psi$-Caputo fractional derivatives, a relationship between the fractional derivative and integral, and various theorems that establish the properties of this operator.