## stamp 2 years ago [CALCULUS III—CHAIN RULE] Find dw/dt by the chain rule where w = xy cos(z) and x = t, y = t^2, and z = arccos(t).

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$w=xy\ cos(z)$$w=t(t^2)\ cos(arccos(t))$$w=t^3t$$w=t^4$$dw/dt=4t^3$ Is that all there is to it or am I mistaken?

2. amistre64

by the chain rule eh $w=xy\ cos(z)$ $w(t)=x(t)y(t)\ cos(z(t))$ $w'(t)=x'(t)y(t)\ cos(z(t))+x(t)y'(t)\ cos(z(t))+x(t)y(t)\ cos'(z(t))$ $w'(t)=x'(t)y(t)\ cos(z(t))+x(t)y'(t)\ cos(z(t))-x(t)y(t)z'(t)\ sin(z(t))$

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So did I goof?

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Or would all that simplify to 4t^3

5. amistre64

we can ignore the (t) parts to clean it up $w'=x'y\ cos(z)+xy'\ cos(z)-xyz'\ sin(z)$ $w'=cos(z)(x'y+xy')-xyz'\ sin(z)$ $w'=cos(z)(x'y+xy')-xyz'\ sin(z)$ x = t, y = t^2, and z = arccos(t) x = 1, y = 2t, and z = $$\frac{1}{\sqrt{1+...}}$$ $w'=cos(cos^{-1}z)(y+2t^2)-t^3z'\ sin(cos^{-1}z)$

6. amistre64

$w'=z(t^2+2t^2)-t^3z'\ sin(cos^{-1}z)$ $w'=3t^2z-t^3z'\ sin(cos^{-1}z)$ is that looking right so far?

7. stamp

You forgot your x' and y' and z' on your derivatives but it is ok because I recognize what you did.

8. amistre64

it does simplify out to 4t^3

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$w′=cos(z)(x′y+xy′)−xyz′ sin(z)$ x'y = t^2 xy' = 2t x'y + xy' = 2t + t^2 = t(2+t) cos(z) = cos(arccos t) = t t( t(2+t) ) - xyz' sin(z)

10. amistre64

provided t not equal -1 or 1 due to that z'

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I am not quite seeing the simplification to 4t^3

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I understand the first part where you showed the formal chain rule of w(t) = x(t) y(t) cos(z(t)) The simplification of it to 4t^3 is still eluding me.

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ps thank you for your assistance so far

14. amistre64

i got lost, let me go back to the w' chains :) and sort it back out

15. stamp

ok

16. amistre64

$w'=x'y\cos(z)+xy'\cos(z)+xy~\cos'(z(t))$ $w'=x'y\cos(z)+xy'\cos(z)-xyz'~\sin(z)$ $w'=x'y\cos(z)+xy'\cos(z)-xyz'~\sin(z)$ $x = t,~ y = t^2, ~z = cos^{-1}(t)$$x' = 1, ~y' = 2t, ~z' =(1-t^2)^{(-1/2)}$ $w'=t^2t+t~2t~t-tt^2(1-t^2)^{(-1/2)}~\sin(\cos^{-1}(t))$ $w'=t^3+2t^3-t^3(1-t^2)^{(-1/2)}~\sin(\cos^{-1}(t))$ $w'=t^3(3-(1-t^2)^{(-1/2)}~\sin(\cos^{-1}(t))$ the key here is in definine sin(cos^-1 (t)) better

17. amistre64

|dw:1361983895329:dw|

18. amistre64

we still have to include the caveat of t not equal to -1 or 1

19. stamp

sin(arccos(t)) = sqrt( 1 - t^2 )

20. amistre64

so that reduces to w' = (3-1) t^3 which means i prolly missed a sign in there someplace along the way :)

21. amistre64

z' = -(1-t^2)^1/2

22. amistre64

i forgot the - part :)

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w' = (3-1)t^3 w' = 2t^3 ?? :(

24. amistre64

check out z', its spose to be negative ... not positive

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Ok. I will ponder all of this and do some work on paper to verify everything. Thank you again for your time.

26. amistre64

youre welcome

27. stamp

@amistre64 Here is the derivative of x(t) y(t) cos z(t) http://www.wolframalpha.com/input/?i=derivative+of+x%28t%29y%28t%29+cos%28z%28t%29%29+

28. amistre64

yes, but z' is negative so that changes the sign .... and carries it to the end as (3+1) not (3-1)