How to find the derivative of ln x?

Question

Tommy Kerr · Accepted Answer

Consider f(x) = lnx and using the first principles work out

f' (x) = lim h \to 0 \frac{f (x + h) - f (x)}{h}

=

lim h \to 0 \frac{\ln (x + h) - \ln x}{h}

=

lim h \to 0 \frac{\ln (\frac{x + h}{x})}{h}

=

lim h \to 0 \frac{\ln (1 + \frac{h}{x})}{h}

=

lim h \to 0 \frac{1}{x} \frac{\ln (1 + \frac{h}{x})}{\frac{h}{x}}

=

\frac{1}{x}

lim h \to 0 \frac{\ln (1 + \frac{h}{x})}{\frac{h}{x}}

=

\frac{1}{x}

, because

[lim h \to 0 \frac{\ln (1 + \frac{h}{x})}{\frac{h}{x}} = 1]

kancsalrma · Accepted Answer

The &quot;how&quot; depends on the definition you are using for ln x.            ln      x        =                  ∫        1        x                    1      t              d      t       and the result is immediate useing the Fundamental Theorem of Calculus.If you start with a definition of             e      x       and find its derivative, then you probably defined ln x by:      y    =          ln      x             ⇔         x    =          e      y      Differentiate implicitely:            d                        d          x                            (      x      )        =          d                        d          x                            (              e        y            )            1    =          e      y                      d        y                    d        x            Hence,                     d        y                    d        x              =          1              e        y              =          1      x

How do you find the derivative of ln x?

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