How Do You Differentiate Trigonometric Functions?

[domyy]

Homework Statement

1. f(x) = 5 sin (8∏x)

2. g(x) = 4∏ [ cos (3∏x) sin (3∏x)]

3. h(x) = cos [sec (5∏x)]

4. Sketch the graph of each function on the indicated interval, making use of relative extrema and points of inflection.

f(x) = 2sinx + sin2x ; [0,2∏]

The Attempt at a Solution

1. f(x) = 5 sin (8∏x)

f'(x) = 5 cos (8∏x) . (8∏)

f'(x) = 40∏ cos (8∏x)

2. g(x) = 4∏ [ cos (3∏x) sin (3∏x)]

g'(x) = 4∏ {[cos (3∏x)][(sin (3∏x)]' + [sin (3∏x)][cos (3∏x)]'}

g'(x) = 4∏ {[cos (3∏x)][cos (3∏x) . 3∏ ] + [ sin (3∏x)][ -sin (3∏x) . 3∏ ]}

g'(x) = 12∏ cos² (3∏x) -12∏ sin² (3∏x)

3. h(x) = cos [sec(5∏x)]

h'(x) = -sin [ sec(5∏x) . 5∏][ tan (5∏x) . 5∏]

h'(x) = -25∏ sec (5∏x) tan(5∏x)

4. Sketch the graph of each function on the indicated interval, making use of relative extrema and points of inflection.

f(x) = 2sinx + sin2x

f'(x) = 2cosx + cos2x . 2

f'(x) = 2cosx + 2cos(2x)

=> f'(x) = 0

2cosx + 2cos(2x) = 0

Now, how to proceed from here?

 

[tiny-tim]

hi domyy!

1 is ok

2 is ok, except haven't you lost one of the ∏s ?

3 is almost completely wrong, have another go, writing it out more carefully at each step

4 now you need to use one of the standard trigonometric identities (cosA - cosB), all of which you should learn

 

[domyy]

:shy: Hi!

3. h(x) = cos [sec(5∏x)]

h'(x) = -sin [ sec(5∏x)][ tan (5∏x)] . 5∏

h'(x) = -5∏sin [ sec (5∏x) tan(5∏x)]

How about now?

What's the wrong with ∏ on nr 2 ? :/

 

[domyy]

I received a warning for excessive use of colors ??

I thought it was actually better for whoever was reading..to separate each problem by color since I was posting more than one.

I didn't do it because I was trying to decorate my post.

My bad.

 

[tiny-tim]

hi domyy!

(btw, i didn't complain about the colours, but i didn't like them)

3 is now ok, except the derivative of sec is tan²

(and, for 2, 4∏*3∏ = 12∏² )

 

[domyy]

Yes. Got it!
=)

Now, I'm going to work on nr. 4