Question

derivative of y=sin^5 (cos^6 x)

Answer 1

Apply the chain rule to solve this problem let u = sin(cos^6(x))

Answer 2

y=u^5, dy/du = 5u^4

Answer 3

\[5\sin^4(\cos^6(x))(\frac{d}{dx}\sin(\cos^6(x))\]

Answer 4

do u get this part so far?

Answer 5

we will be applying the chain rule several times...

Answer 6

I'm feeling confident! I am getting that so far :)

Answer 7

I think I may have an answer.

Answer 8

now we are going to make another substitution to make a reduction on the power of the trig implicit part
\[v = \sin^6(x)\]
\[\frac{dsin(v)}{dv}=\cos(v)\]

Answer 9

Ok, I didn't do the substitution there.

Answer 10

\[5\sin^4(\cos^6(x))((\cos(\cos^6))(\frac{d}{dx}\cos^6(x)))\]

Answer 11

so the final substituiton we are going to make would be

Answer 12

m = cos(x) , (du/dx)^6=6m^5

Answer 13

\[5\cos(\cos^6(x))\sin^4(\cos^6(x))(6\cos^5(x)(\frac{d}{dx}\cos(x)))\]

Answer 14

fnaly would you like to do the last step?

Answer 15

\[-30\cos(\cos^6(x))\sin^4(\cos^6(x))(\cos^5(x)(\sin(x)))\]

Answer 16

OK, I think I have it.

Answer 17

That is what I got other than I forgot I could move the - from the last step from -sin to the front. Wow! Thanks!! Lot's of practice with the chain rule :)

Answer 18

you're welcome...