Question

How do you differentiate `y = (sqrt(cos x))/(5lnx)`?

Answer 1

`-(xln(x)sin(x)+2cos(x))/(10xsqrt(cos(x))(ln(x))^2)`

Explanation:

use quotient rule:

`d/dx(f(x)/g(x))=(f'(x)g(x)-g'(x)f(x))/(g(x))^2`

(see this proof or this)

using above format for this question, `f(x)=sqrt(cosx)` and `g(x)=5ln(x)`

`y'(x)=(d/dx(sqrt(cos(x)))(5ln(x))-d/dx(5ln(x))sqrt(cos(x)))/(5ln(x))^2`

`y'(x)=(1/(2sqrt(cos(x)))*d/dx(cos(x))(5ln(x))-(5/x)sqrt(cos(x)))/(5ln(x))^2`

`y'(x)=(1/(2sqrt(cos(x)))*(-sin(x))(5ln(x))-(5/x)sqrt(cos(x)))/(5ln(x))^2`

when you simplify you should get something like:

`-(xln(x)sin(x)+2cos(x))/(10xsqrt(cos(x))(ln(x))^2)`