OpenStudy (anonymous):

calculate the derivative and simplify h(x)=3xsin^2 (x) + cos^3 (x)

3 years ago
OpenStudy (science0229):

Let's start with the first term.

3 years ago
OpenStudy (science0229):

How do differentiate 3xsin^2 (x)?

3 years ago
OpenStudy (anonymous):

product rule?

3 years ago
OpenStudy (science0229):

Correct

3 years ago
OpenStudy (anonymous):

d/dx(3x)sin(x)+3xd/dx(sinx)

3 years ago
OpenStudy (science0229):

good, so what do you get if simplified?

3 years ago
OpenStudy (anonymous):

3sin^2(x)+3cos^2(x)

3 years ago
OpenStudy (science0229):

wait. Sorry. You're not quite there.

3 years ago
OpenStudy (science0229):

\[\frac{ d }{ dx }3xsin^2x=(\frac{ d }{ dx }3x)\sin^2x+3x(\frac{ d }{ dx }\sin^2x)\]

3 years ago
OpenStudy (science0229):

\[=3\sin^2x+3x(2\sin x \cos x)=3\sin^2x+6x \sin x \cos x\]

3 years ago
OpenStudy (science0229):

Do you know the formula for differentiating composite function?

3 years ago
OpenStudy (anonymous):

is that the chain rule?

3 years ago
OpenStudy (science0229):

Correct. I think you forgot to use it when differentiating sin^2x ;)

3 years ago
OpenStudy (anonymous):

Yeah i get confused when it needs to be used. Does it always have to used with trig functions?

3 years ago
OpenStudy (science0229):

no. You always use it when differentiating powers of a function. So for y={f(x)}^n, chain rule is always used.

3 years ago
OpenStudy (anonymous):

Ok yeah that makes sense.

3 years ago
OpenStudy (science0229):

Now, using that, try differentiating the second term of the original problem.

3 years ago
OpenStudy (anonymous):

So would d/dx cos^3(x) be 3cos(x)*-sin(x)

3 years ago
OpenStudy (science0229):

Close. y={f(x)}^n y'=n{f(x)}^(n-1)*f'(x)

3 years ago
OpenStudy (anonymous):

3cos^2(x)-sin(x)

3 years ago
OpenStudy (science0229):

correct. Now add those 2 terms.

3 years ago
OpenStudy (anonymous):

-3cos^2(x)sin(x) is that what you are meaning?

3 years ago
OpenStudy (science0229):

Remember how we differentiated the first term? Add that and the one you just found.

3 years ago
OpenStudy (anonymous):

k i wasn't sure which one you meant 3sin^2(x)+3x(sin(x)cos(x)-3cos^2(x)sin(x)

3 years ago
OpenStudy (science0229):

What was the answer for the differentiation of the first term?

3 years ago
OpenStudy (anonymous):

wasn't it 3sin^2(x)+3x(2sin(x)cos(x))

3 years ago
OpenStudy (science0229):

Oh. I see. In my personal opinion, I think you should write it as 3sin^2(x)+6xsin(x)cos(x), but it's your choice.

3 years ago
OpenStudy (science0229):

Now that you got it, add it with your 2nd term. What do you get?

3 years ago
OpenStudy (anonymous):

3sin^2(x)+6xsin(x)cos(x)-3cos^2(x)sin(x) would the next step be to factor out a 3?

3 years ago
OpenStudy (science0229):

Actually, you can just leave it like that. Say hi to your answer :)

3 years ago
OpenStudy (anonymous):

It says i have to simplify?

3 years ago
OpenStudy (anonymous):

is that as simplified as it can get?

3 years ago
OpenStudy (science0229):

Yep!

3 years ago