Question

How do I find the derivative of `f(x)= (ln10x) / (x+4)`?

Answer 1

`f'(x) = 1/(x+4) (1/x - ln(10x)/(x+4))`

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}`