how do we know the derivative of ln(x) is 1/x (the definition & implicit differentiation)

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Why Is the Derivative of ln(x) Equal to 1/x? | Calculus Proof ExplainedWhy Is the Derivative of ln(x) Equal to 1/x? | Calculus Proof Explained
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Why is the derivative of lnx equal to 1/x?Why is the derivative of lnx equal to 1/x?
If y=lnx, dy/dx=1/x (Proof - Alternative Short Cut, Implicit Differentiation)If y=lnx, dy/dx=1/x (Proof - Alternative Short Cut, Implicit Differentiation)
The Derivative of ln(x) via Implicit DifferentiationThe Derivative of ln(x) via Implicit Differentiation
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Proof: the derivative of ln(x) is 1/x | Advanced derivatives | AP Calculus AB | Khan AcademyProof: the derivative of ln(x) is 1/x | Advanced derivatives | AP Calculus AB | Khan Academy
derivative of ln(1/x) vs derivative of 1/ln(x)derivative of ln(1/x) vs derivative of 1/ln(x)