## anonymous 5 years ago find the derivative of sin^4 (x)

1. anonymous

would it be cos^4x?

2. anonymous

i no that the derivative of sin is cos, but the power of 4 confuses me

3. anonymous

you can split it up sin^2x * sin^2x than do power reducing

4. anonymous

$\frac{d}{dx}\sin^2 x=4 \sin ^3 x .\cos x$

5. anonymous

sin^4 x sorry

6. anonymous

Have you learned about the chain rule yet?

7. anonymous

Rewrite it as (cosx)^4

8. anonymous

ya i learned the chain rule

9. anonymous

Oops (sinx)^4

10. anonymous

Just like newone wrote and chain rule it.

11. anonymous

so it would be 4sinx^3 * cos?

12. anonymous

yes

13. anonymous

ok so what happens here is this. you have to use chain rule. soo.... sin^4(x) --> 4sin^3(x)cos(x)

14. anonymous

you can even check it out in wolframalpha

15. anonymous

so how do i no when to use the chainrule?

16. myininaya

when you have a function inside a function

17. anonymous

so for this question sin is the inside function and 4 is the outside?

18. myininaya

so we have f(g(x))=(sinx)^4 where f(x)=x^4 and g(x)=sinx

19. anonymous

ooooo i c!!

20. myininaya

the chain rule is this f'(g(x))*g'(x)

21. anonymous

ok. think about it like this. when you find derivative of x^2 its 2x. we stop right here becuase we note that x is not in any function. say if we have cos(2x) then we have to use chain rule. becuase x is inside another function. so derivative of cos(2x) = -sin(2x) = so we derivce the function first. and then we note that we have to further derive x. so we derivce 2x = 2. so we get the final answer cos(2x) = -2sin(2x) chain rule states:

22. anonymous

thank u guys!

23. anonymous

Let $u = (sin\ x) \rightarrow (sin\ x)^4 = u^4\ and\ du/dx = cos\ x$ Following the chain rule we have $d/dx(u^4) = d/du(u^4) * du/dx= 4u^3 * (\cos\ x) = 4(\sin\ x)^3(\cos\ x)$