Derivative of trigonometric functions

[domyy]

Homework Statement

g(x) = 4∏ [cos(3∏x) sin (3∏x)]

The Attempt at a Solution

g(x) = 4∏ [cos(3∏x) sin (3∏x)]'

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

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

Now, my question is: Can I combine the numbers and have the answer as:

- 36∏ [cos²(3∏x) + sin²(3∏x)]

Thank you so much!

 

[Dick]

Homework Statement

g(x) = 4∏ [cos(3∏x) sin (3∏x)]

The Attempt at a Solution

g(x) = 4∏ [cos(3∏x) sin (3∏x)]'

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

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

Now, my question is: Can I combine the numbers and have the answer as:

- 36∏ [cos²(3∏x) + sin²(3∏x)]

Thank you so much!

The -36 is certainly wrong and isn't it a difference between the cos^2 and sin^2 parts? Why would you think you can do that? Show how you combined. Use algebra.

 

[domyy]

Oh yes. I was thinking of transferring the negative sign from sin to 3∏.

How about this final answer:

-12∏ sin²(3∏x) + 12∏cos²(3∏x)

 

[Dick]

That's much better.

 

[domyy]

Thank you SO MUCH!

 

[domyy]

Oh I have a question:

In the following problem

5 sin (8∏x)

They isolated the 5 to then find the derivative of sin (8∏x)

meaning they are not taking the derivative of 5, right? because taking its derivative would

result in zero.

Now, for the problem

cos (sec (5∏x))

I believe I am supposed to take the derivate of cos.

How do I know when I am supposed to take the derivative of these numbers?

 

[Dick]

Oh I have a question:

In the following problem

5 sin (8∏x)

They isolated the 5 to then find the derivative of sin (8∏x)

meaning they are not taking the derivative of 5, right? because taking its derivative would

result in zero.

Now, for the problem

cos (sec (5∏x))

I believe I am supposed to take the derivate of cos.

How do I know when I am supposed to take the derivative of these numbers?

You are probably overcomplicating this. Taking the the derivative of numbers is never a problem, the derivatives of them are zero. The real problem is with the parts that are functions of x. Your last problem needs to use the chain rule. Look it up if you don't know it.

 

[HallsofIvy]

Oh I have a question:

In the following problem

5 sin (8∏x)

They isolated the 5 to then find the derivative of sin (8∏x)

meaning they are not taking the derivative of 5, right? because taking its derivative would

result in zero.

One of the first things you should have learned in Calculus is that "the derivative of C times f(x) (C is a constant) is C times the derivative of f".

Now, for the problem

cos (sec (5∏x))

I believe I am supposed to take the derivate of cos.

How do I know when I am supposed to take the derivative of these numbers?

You don't "take the derivative of numbers" because, as you said before, the derivative of a constant (number) is 0. You differentiate the functions by using the "chain rule".