Question

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

Answer 1

Let's start with the first term.

Answer 2

How do differentiate 3xsin^2 (x)?

Answer 3

product rule?

Answer 4

Correct

Answer 5

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

Answer 6

good, so what do you get if simplified?

Answer 7

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

Answer 8

wait. Sorry.
You're not quite there.

Answer 9

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

Answer 10

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

Answer 11

Do you know the formula for differentiating composite function?

Answer 12

is that the chain rule?

Answer 13

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

Answer 14

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

Answer 15

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

Answer 16

Ok yeah that makes sense.

Answer 17

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

Answer 18

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

Answer 19

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

Answer 20

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

Answer 21

correct.
Now add those 2 terms.

Answer 22

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

Answer 23

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

Answer 24

k i wasn't sure which one you meant

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

Answer 25

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

Answer 26

wasn't it

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

Answer 27

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.

Answer 28

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

Answer 29

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

Answer 30

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

Answer 31

It says i have to simplify?

Answer 32

is that as simplified as it can get?

Answer 33

Yep!