Question

What is the derivative of (x^2)(sinx)(tanx) without using the chain rule?

Answer 1

You can manipulate your function remembering that:
`tan(x)=sin(x)/cos(x)`
and `sin^2(x)=1-cos^2(x)`
and get:

from this point onwards it is a piece of cake!!!
:-)

Answer 2

Use the product rule for three factors:

`d/(dx)(fgh) = f'gh+fg'h+fgh'`

For `y = x^2sinxtanx`, we get

`y' = 2xsinxtanx + x^2 cosxtanx+x^2sinxsec^2x`

We ca rewrite the middle term more simply, and we may choose to rewrite the third term:

`y' = 2xsinxtanx + x^2 sinx+x^2tanxsecx`.

`d/(dx)(fgh) =f'[gh]+f d/(dx)[gh]`

`color(white)"sssssssss"` `=f'gh+f[g'h+gh']`

`color(white)"sssssssss"` `=f'gh+fg'h+fgh'`.