Question

Use the definition of the derivative to differentiate f(x)= In x

Answer 1

By def. of the derivative, we have for y = ln(x),

$\displaystyle \frac{dy}{dx} = \lim_{h\to0} \frac{\ln(x+h)-\ln(x)}{h}$

$\displaystyle \frac{dy}{dx} = \lim_{h\to0} \frac1h \ln\left(\frac{x+h}{x}\right)$

$\displaystyle \frac{dy}{dx} = \lim_{h\to0} \ln\left(1+\frac hx\right)^{\frac1h}$

Substitute y = h/x, so that as h approaches 0, so does y. We then rewrite the limit as

$\displaystyle \frac{dy}{dx} = \lim_{y\to0} \ln\left(1+y\right)^{\frac1{xy}}$

$\displaystyle \frac{dy}{dx} = \frac1x \lim_{y\to0} \ln\left(1+y\right)^{\frac1y}$

Recall that the constant e is defined by the limit,

$\displaystyle e = \lim_{y\to0} \left(1+y\right)^{\frac1y}$

Then in our limit, we end up with

$\displaystyle \frac{dy}{dx} = \frac1x \ln(e) = \boxed{\frac1x}$

In Mathematica, use

D[Log[x], x]