100 POINTS
Evaluate the derivative of the function number 38

 1/ sqrt(1+ln^2(x))    * 1/(ln^2x +1) * 1/x

Step-by-step explanation:

f(x) = sin (tan^-1 (ln(x)))  

u substitution

       d/du (sin u)  * du /dx

            cos (u) * du/dx

Let u =(tan^-1 (ln(x)))     du/dx =d/dx  (tan^-1 (ln(x)))

v substitution

                                             Let v = ln x                dv/dx = 1/x

                                              d/dv  (tan ^-1 v)  dv/dx

                                                          1/( v^2+1) * dv/dx

                                                        =1/(ln^2x +1) * 1/x

Substituting this back in for  du/dx

cos (tan^-1 (ln(x))  * 1/(ln^2x +1) * 1/x

We know that cos (tan^-1 (a)) = 1/ sqrt(1+a^2)

cos (tan^-1 (ln(x))  * 1/(ln^2x +1) * 1/x

 1/ sqrt(1+ln^2(x))     * 1/(ln^2x +1) * 1/x

cos((tan^-1)(lnx))/[x(1 + ln²x)]

Step-by-step explanation:

Sin((tan^-1)(lnx))

First differentiate the outer function 'sin'

Then inner functions

derivative of:

sin(x) is cos(x)

tan^-(x) = 1/(1+x²)

lnx = 1/x

cos((tan^-1)(lnx)) × [1/(1 + ln²x)] × 1/x

cos((tan^-1)(lnx))/[x(1 + ln²x)]

option a is correct.

step-by-step explanation:

scale factor: it is used to find proportional measurements.

proportions means having the same ratio.

scale factor is the ratio of the model measurement to the actual measurement in a simplest form.

or we can say that the ratio of any two corresponding lengths in two similar geometric figures

if the scale factor is more than 1, then it is an enlargement,

if the scale factor less than 1, then it is a reduction.

given the two geometric figure:

by definition of proportions, we have