## lilly21 3 years ago Calculus HELP! find the derivative of the function. g(t)=5cos^3(times pie)(times t)

1. zepp

$$\large g(t) = 5cos^3(\pi t)$$

2. ksaimouli

HINT - pie and t are constant

3. ksaimouli

use the trig formulas for derivatives

4. lilly21

ok zepp so tell me if i have it right so far..... y(prime)=u(prime) times v+v(prime) times u = no wait.... im stuck

5. lilly21

where do i take the derivative first for this function?

6. zepdrix

Take the derivative of the OUTERMOST function first. Trig functions can be a little tricky to read since the power is written in a different location. Identify the outermost function. $5(\cos(\pi t))^3$ That is another way it can be written so you can easily identify the outermost function.

7. UnkleRhaukus

use the chain rule , not the product rule

8. lilly21

ok so whta rule would i be using in this case when i take ther derivative of the outermost function

9. zepdrix

The power rule :) and then as Unkle mentioned, the chain rule after that.

10. lilly21

ok..... so using the power rule in order to find the outermost function first this would be 15?

11. zepdrix

$15(\cos(\pi t))^2 * (d/dt)(\cos(\pi t))$ good you got the first step :) by the chain rule, you must now multiply by the derivative of the inside, as I have written here.

12. UnkleRhaukus

$(f\circ h)'(x)=f(h(x))'=f'(h(x))\cdot h'(x)$

13. lilly21

ok i got this step now i must find the derivative of (cos(piet))^2 right?

14. zepdrix

Don't look at the square anymore. You already dealt with that outermost function. the inner function is cos(pi t)

15. lilly21

ok so i find the derivative of that and that is -sin(piet) right?

16. zepdrix

$15(\cos(\pi t))^2*(-\sin(\pi t))*(d/dt)(\pi t)$ Good! now we have one last step. there is an inner inner function that you need to differentiate still, as I have written above.

17. lilly21

ok! thanks! ummm... would this be no wait u lost me

18. zepdrix

The innermost function is (pi t). So you need to multiply by the derivative of that :D Find the derivative! :D

19. lilly21

u see i thought of that being the answer but i wasn't quite sure... umm okay so the derivative of piet is just piet?

20. zepdrix

$(d/dt)(\pi t)=\pi(d/dt)t=?$

21. zepp

$$\pi$$ is a constant, and when a constant combined to a variable, the variable goes to 1 but the constant stay.

22. lilly21

zepdrix :[

23. lilly21

im sorry u guys lost me completely

24. zepp

Let's restart everything, haha

25. zepdrix

nooo don't get lost :D you were so close!

26. lilly21

is it just pie?

27. zepdrix

yayyyy \:D/

28. zepdrix

pi* lol

29. UnkleRhaukus

$g(t) = 5\cos^3(\pi t)$ $f(x)=5\cos^3(x)$ $h(x)= \pi x$

30. rainbow22

CHAIN RULE :D

31. zepp

Okay, $\large g(t) = 5cos^3(\pi t)$We identified $$5cos(\pi t)$$ as the inner function, and $$(Stuffshere)^3$$ as the outter function, so we could apply the power rule here and we apply the chain rule $3*5(cos^2(\pi t))*\frac{d}{dt}cos^2(\pi t)*\frac{d}{dt}(\pi t)$

32. lilly21

ok how it pi though

33. lilly21

how is it pi

34. zepp

$$\large 3*5(cos^2(\pi t))*\frac{d}{dt}cos(\pi t)*\frac{d}{dt}(\pi t)$$*** Sorry

35. zepdrix

what is the derivative of 2x? its just 2 times the derivative of x. So you apply the power rule, (derivative of x^1 = 1.. so you end up with 2. :O yes? same thing is happening here. don't let the pi confuse you, its just a number

36. lilly21

ok !!!! i think i got!!! THANKS SOOOOO MUCH I WAS HAVING TROUBLEE!!!!!!