Question

Question #44491

Answer 1

This is a product of three functions.

Explanation:

I use the product rule in the order:

`d/dx(FS) = F'S+FS"`

(The derivative of a product is the derivative of the first times the second, plus the first times the derivative of the second.)

For three functions, we have

`f(x) = uvw = u(vw)`

`f'(x) = u'(vw)+u(vw)'`

`= u'vw+u(v'w+vw')`

`= u'vw+uv'w+uvw'`

For four functions

`d/dx(tuvw) = t'uvw+tu'vw+tuv'w+tuvw'`

In this question `f(theta) = theta cos theta sin theta`

so,

`f'(theta) = 1 cos theta sin theta +theta(-sintheta sintheta + costhetacostheta)`

`= cos theta sin theta + theta( cos^2 theta - sin^2 theta)`

The quantity in parentheses is equal to `cos 2theta`, so we can write

`f'(theta) = cos theta sin theta + theta cos2 theta`