Question

Hi. I need help with these questions (see image)

Please show workings.

Question : Find the derivative with respect to x of each of the following:
(see image)​

Answer 1

see explanation

Explanation:

Note that
`log_(e)` x = lnx ( natural logarithm)

Using the standard derivatives

`(d)/(dx)` (lnx ) =
`(1)/(x)` ,
`(d)/(dx)` (
`log_(a)` x ) =
`(1)/(xlna)`

Using the chain rule

Given y = f(g(x)) , then

`(dy)/(dx)` = f'(g(x))) × g'(x) ← chain rule

(c)

y =
`log_(10)` (x² - 2) then

`(dy)/(dx)` =
`(1)/((x^2-2)ln10)` ×
`(d)/(dx)` (x² - 2)

=
`(1)/((x^2-2)ln10)` × 2x

=
`(2x)/((x^2-2)ln10)`

(d)

y = ln(
`(1)/(x)` )

`(dy)/(dx)` =
`(1)/((1)/(x) )` ×
`(d)/(dx)` (
`x^(-1)` )

= x × -
`x^(-2)`

= x ×
`(-1)/(x^2)`

= -
`(1)/(x)`