Use Mathematica to verify.

Question

Use the definition of the derivative to differentiate

f (x) = ln x.

Answer 1

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Okay to solve this problem, the definition of derivative is basically in terms of limits:

f'(x) = lim _h→0 (f(x+h) - f(x)) / h

We are also going to need the definition of e in terms of limits as x approaches 0:

e = lim_x→0(1 + x)^1/x

and ln(e) = 1

For the derivative of f(x) = ln (x), we also have f(x + h) = ln (x + h)

f'(x) = lim _h→0 (ln(x+h) - ln(x)) / h

Using this property natural logarithm, ln(a) - ln (b) = ln (a/b):

f'(x) = lim _h→0 (ln((x+h)/x) ) / h = (1/h) lim _h→0 (1/h)(ln((x+h)/x))

Using this property of natural logarithm, a ln(b) = ln (b^a)

f'(x) = lim _h→0 ln(((x+h)/x)^1/h)= lim _h→0 ln((1 + h/x)^1/h)

Let n = h/x, then as n→0, so as h→0 and h = nx

Substitute:

f'(x) = lim _n→0 ln(1 + n)^1/(nx) = lim _n→0 ln(1 + n)^(1/n)(1/x)

Using this property again, a ln(b) = ln (b^a)

f'(x) = (1/x) lim _n→0 ln(1 + n)^(1/n)

Using the definition of e

f'(x) = (1/x) ln(e)

f'(x) = (1/x)