Question

plz answer this question fast 5 question plz

Answer 1

Answer:

f' ( 2π / 3 ) = ( 1 - √3 ) / 2

Hope this helps you out.  All the best!!!!

Step-by-step explanation:

Things we need to know:

1. (d/dx) sin x = cos x
2. (d/dx) cos x = - sin x
3. cos ( 2π / 3 ) = - 1 / 2
4. sin ( 2π / 3 ) = √3 / 2

The danger in 1 and 2 is getting the signs wrong.  Either commit them to memory, or remember the functions visually and interpret the derivatives in your picture.

For 3 and 4, if you don't keep these things in your memory, that's fine.  Just practice working them out when you need them.  For instance, 2π is all the way around a circle (starting at the x-axis and going anticlockwise), so π is half way around and 2π / 3 is two thirds of that.  If you prefer degrees, you might just say 2π corresponds to 360°, so 2π / 3 corresponds to 360° / 3 =  120°.  Draw your diagram and work it out from there.  The diagram involves half of an equilateral triangle and should be familiar to you.

Now to the task.

Given f ( x ) = cos x - sin x

We want f' ( 2π / 3 )

First, facts 1 and 2 above, we have

f' ( x ) = - sin x - cos x

Substituting x = 2π / 3,

f' ( 2π / 3 )

= - sin (2π/3) - cos (2π/3)

= - √3 / 2 - ( - 1 / 2 )

= 1 / 2  -  √3 / 2

= ( 1 - √3 ) / 2