Question

Question #de1a0

Answer 1

Substitution and chain rule

Explanation:

Let `theta =3x^2` and `p = cos theta` then

`y = sin^3(cos(3x^2))`
`y = sin^3p`

For `y'` we can use chain rule as

`(dy)/(dx) = (dy)/(dp)(dp)/(d theta)(d theta)/(dx)`

`(dy)/(dp) = 3sin^2pcosp`

`(dp)/(d theta) = -sin theta`

`(d theta)/dx = 6x`

`(dy)/(dx) = 3sin^2pcosp\times(-sin theta)\times6x`

`= -18xsin^2(cos(3x^2))cos(cos(3x^2))\times sin(3x^2)`
`=-9sin(2cos(3x^2))sin(cos(3x^2)) sin(3x^2)`