Differentiating Logarithmic Functions with Base e

Key Questions

  • What is the derivative of f(x)=ln(x)/xf(x)=ln(x)x ?

    By Quotient Rule,

    y'={1/x cdot x-lnx cdot 1}/{x^2}={1-lnx}/{x^2}

    This problem can also be solved by the Product Rule

    y'=f'(x)g(x)+f(x)g(x)

    The original function can also be rewritten using negative exponents.

    f(x)=ln(x)/x=ln(x)*x^-1

    f'(x)=1/x*x^-1+ln(x)*-1x^-2

    f'(x)=1/x*1/x+ln(x)*-1/x^2

    f'(x)=1/x^2-ln(x)/x^2

    f'(x)=(1-ln(x))/x^2

    Wataru · 3 · Sep 22 2014
  • What is the derivative of f(x)=ln(g(x)) ?

    The answer would be f'(x) = 1/g(x)*g'(x) or it can be written as f'(x)=(g'(x))/g(x).

    To solve this derivative you will need to follow the chain rule which states:

    F(x)=f(g(x)) then F'(x)=f'(g(x))*g'(x)

    Or without the equation, it the derivative of the outside(without changing the inside), times the derivative of the outside.

    The derivative of h(x) = ln(x) is h'(x) = 1/x.

    Michael B. · 3 · Sep 24 2014
  • What is the derivative of log_e(x)?

    Answer:

    1/x

    Explanation:

    log_e(x) is commonly denoted as ln(x), the natural log.

    =>d/(dx) ln(x) = 1/x

    If you would like a proof, we can derive it from the limit definition:

    lim_(delta x->0)(f(x+delta x)-f(x))/(delta x)

    = lim_(delta x->0)(ln(x+delta x)-ln(x))/(delta x)

    = lim_(delta x->0)(ln((x+delta x)/(x)))/(delta x)

    = lim_(delta x->0)1/(delta x)ln(1+(delta x)/x)

    = lim_(delta x->0)ln((1+(delta x)/x)^(1/(delta x)))

    = lim_(delta x->0)ln((1+(delta x)/x)^(1/(delta x)))

    "Let " tau equiv (delta x)/x:

    = lim_(delta tau->0)ln((1+tau)^(1/(xtau)))

    = lim_(delta tau->0)ln[((1+tau)^(1/(tau)))^(1/x)]

    = ln[(e)^(1/x)]

    = 1/x ln(e)

    = 1/x (1)

    = 1/x

    NJ · 1 · Apr 4 2018

Questions

Differentiating Logarithmic Functions