How do I find the derivative of #f(x)= (ln10x) / (x+4)#?
1 Answer
Mar 13, 2016
Explanation:
Use the quotient rule.
#frac{"d"}{"d"x}(u/v) = frac{v frac{"d"u}{"d"x} - u frac{"d"v}{"d"x}}{v^2}#
In this question
#u = ln(10x) = ln(10) + ln(x)#
#frac{"d"u}{"d"x} = frac{"d"}{"d"x}(ln(10)) + frac{"d"}{"d"x}(ln(x))#
#= 1/x#
#v = x + 4#
#frac{"d"v}{"d"x} = 1#
So, plugging it all in,
#f'(x) = frac{v frac{"d"u}{"d"x} - u frac{"d"v}{"d"x}}{v^2} #
#= frac{(x + 4) (1/x) - ln(10x) (1)}{(x + 4)^2}#
#= 1/(x+4) (1/x - ln(10x)/(x+4))#
#= frac{x + 4 - xln(10x)}{x(x + 4)^2}#
