Place the argument of the logarithmic function to be differentiated into the denominator of a fraction under the number one. Example, ln(x - 2) -> 1 / x - 2.
Place the expression into an integral with respect to "x". For example, ∫ 1 / x-2 dx.
Perform a "u-substitution" for the expression in the denominator. For example, x - 2 = u.
Take the derivative of "u." For example, u = x - 2 -> du / dx = 1. The derivative of X - 2 is 1.
Solve the differential equation for dx. For example, du / dx = 1 -> du = dx. Multiply the equation on both sides by dx.
Substitute "u" and "du" into the original expression. Substituting "u" for x - 2 -> 1 / x - 2 = 1 / u and substituting du for dx, then ∫ 1 / d - 2 dx -> ∫ 1 / u du.
Integrate the simplified integral: ∫ 1 / u du = Ln(U).
Substitute the original terms. U was defined as x - 2 earlier, and therefore, Ln(u) = ln(x - 2). This successful u-substitution proves that the anti-derivative of 1 / x - 2 is, in fact, ln(x - 2). Due to their inverse natures, integration and differentiation, this proves that the derivative of ln(x -2 ) is equal to 1 / x - 2. This process works with any logarithmic function.