Question #44491

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Answer 1 · 2017-02-23T20:18:24

This is a product of three functions.

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 #