Solving a derivative: y = x e^ x csc x

[grapz]

y = x e^ x csc x

This is what i did:

y' =( x e ^ x ) ( csc x )

i use product rule two times

y' = ( e ^ x ( x) + e ^ x ) ( csc x )

y' = ( - cot x csc x ) ( e ^ x ( x) + e^x) + ( e^x + e^x (x) + e ^ x) ( cscx)

is this right?

 

[o_O]

Not quite. When you're using the product rule, you have to split y into two expressions and use the rule accordingly (since the product rule deals with two functions). So say that this equation was expressed as y = f(x)g(x). You could let f(x) = xe^x and g(x) = cscx or f(x) = x and g(x) = e^xcscx. Either way, the product rule will work the same and give you equivalent answers for the derivative.

y' = f'(x)g(x) + f(x)g'(x)
y' = (xe^x)'(cscx) + (xe^x)(cscx)'

As you can see, f'(x) requires the additional product rule which you could probably manage yourself.

 

[tkhunny]

Re: Solving a derivative

y = x e^ x csc x

This is what i did:


y' =( x e ^ x ) ( csc x )

i use product rule two times

y' = ( e ^ x ( x) + e ^ x ) ( csc x )

y' = ( - cot x csc x ) ( e ^ x ( x) + e^x) + ( e^x + e^x (x) + e ^ x) ( cscx)

is this right?

You have the right idea, but you have confused yourself. Mostly, you let the notation get out of hand.

y' =( x e ^ x ) ( csc x )

That is just wrong. That it 'y'. Why do you have it listed as "y-prime"? I know you don't mean that, but that is what you wrote.

y' = ( e ^ x ( x) + e ^ x ) ( csc x )

Again with the y-prime as in intermediate result. No good.

Be more careful.

y = x e^ x csc x

Take f(x) = x e^x
Then f'(x) = e ^ x ( x) + e ^ x

You had that, but with bad notation.

y = f(x) csc(x)

By the product rule

y' = f(x)*( - cot x csc x ) + f'(x)*csc(x)

This is were you went really off. Rather than insert the function, f(x) and its derivative, f'(x), you managed to use f'(x) and f"(x). You should have gotten:

y' = (x e^x)*( - cot x csc x ) + (e ^ x ( x) + e ^ x)*csc(x)

This is much simpler that your expression. Also, this can be simplified substantially.

y' = (e^x)(csc(x))(x + 1 - x cot(x))

 

[grapz]

Ahhhhhh. Yes, thank you. I realize my error now . Thanks