Question #44491

1 Answer
Feb 23, 2017

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