derivation of the derivative of ln x using limits d/dx(ln x)= 1/x proof rigorous Calculus AB BC

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In this clip we use the limits definition of derivatives to show that the derivative of the natural logarithm of x is 1/x. some key formulas we used includes the properties of logarithms and the limit definition of derivatives.

The student will define and apply the properties of limits of functions. Limits will be
evaluated graphically and algebraically. This will include
a) limits of a constant;
b) limits of a sum, product, and quotient;
c) one-sided limits; and
d) limits at infinity, infinite limits, and non-existent limits. *
*AP Calculus BC will include l'Hopital's Rule, which will be used to find the limit of
functions whose limits yield the indeterminate forms: 0/0 and ∞/∞.
APC.3 The student will use limits to define continuity and determine where a function is
continuous or discontinuous. This will include
a) continuity in terms of limits;
b) continuity at a point and over a closed interval;
c) application of the Intermediate Value Theorem and the Extreme Value Theorem; and
d) geometric understanding and interpretation of continuity and discontinuity.
APC.4 The student will investigate asymptotic and unbounded behavior in functions. This will
include
a) describing and understanding asymptotes in terms of graphical behavior and limits
involving infinity; and
b) comparing relative magnitudes of functions and their rates of change.