Exponential Distribution of Trigonometric Functions?

[justin345]

Homework Statement

e^3x(3sin x-cos x)

The Attempt at a Solution

e^3x3(3sin x-cos x)+(3cos x+sin x)e^3x=10e^3xsin x.

Is that right?

 

[berkeman]

Homework Statement

e^3x(3sin x-cos x)

The Attempt at a Solution

e^3x3(3sin x-cos x)+(3cos x+sin x)e^3x=10e^3xsin x.

Is that right?

I don't think so, at least not the intermediate steps.

Could you try distributing the e term first, and then take the derivative of the two resulting terms, using the chain rule on each?

 

[justin345]

OK, so let me then get rid of parenthesis. I will show you my intermediate steps.

e^3x3sinx-e^3xcosx=
3e^3xsinx+e^3x3cosx-(3e^3xcosx+e^3x(-sinx))=
10e^3xsin x

 

[berkeman]

OK, so let me then get rid of parenthesis. I will show you my intermediate steps.

e^3x3sinx-e^3xcosx=
3e^3xsinx+e^3x3cosx-(3e^3xcosx+e^3x(-sinx))=
10e^3xsin x

I'm not seeing the chain rule being used. Remember, it's the first term multiplied by the derivative of the second term, plus the second term multiplied by the derivative of the first term...

http://en.wikipedia.org/wiki/Chain_rule

.

 

[berkeman]

Also, you should be careful to distinguish between the initial terms, and the derivative that you are solving for. Write it like this:

(e^3x3sinx-e^3xcosx)' =

Or

d/dx(e^3x3sinx-e^3xcosx) =

 

[justin345]

I am sorry, I don't quite understand what you are saying. I performed chain rule to the best of my knowledge. I don't know what else to do.

 

[lurflurf]

Don't worry about berkeman's confusing, unclear, and not helpful post. Your work is quite correct, but it may help to use a few more steps until you have more practice.

product rule
[e^3x(3sin x-cos x)]'=(e^3x)'(3sin x-cos x)+e^3x(3sin x-cos x)'
chain rule and difference rule
=3e^3x(3sin x-cos x)+e^3x(3sin' x-cos' x)
distribute
=e^3x(9sin x-3cos x)+e^3x(3cos x+sin x)
gather like terms (exponential)
=e^3x(9sin x-3cos x+3cos x+sin x)
gather like terms (sine and cosine)
=e^3x(10sin x)
final answer
=10e^3xsin x

 

[berkeman]

Don't worry about berkeman's confusing, unclear, and not helpful post. Your work is quite correct, but it may help to use a few more steps until you have more practice.

Sorry if I was confusing. Should I have been using the term product rule instead of the more general chain rule? It was the intermediate steps that I was trying to get laid out.

 

[Mark44]

Sorry if I was confusing. Should I have been using the term product rule instead of the more general chain rule? It was the intermediate steps that I was trying to get laid out.

Both rules need to be used in this problem. The product rule is used first, because the function is a product - e^3x(3sin x-cos x). Then, to differentiate e^3x, the chain rule is called for.

 

[justin345]

Does it mean that my calculation is correct?

 

[Mark44]

Yes.