Review in Variable Differentiation

[jgreen520]

Please see attached.

I was looking for an explanation of the answer I have attached. Its been a little while and was just looking for the logic behind the differentiation shown for this problem. Its basically an optimization problem where I am looking for the minimum angle (theta) for the least amount of force for T_ab.

Thanks

 

[haruspex]

It uses the standard rule for differentiating a quotient: d(u/v) = (vdu - udv)/v²

 

[HallsofIvy]

You can also do the differentiation of a quotient as the differentiation of a product using the chain rule: $u/v= uv^{-1}$ so that
$$(u/v)'= (uv^{-1})'= u'v^{-1}+ u(v^{-1})'= u'v^{-1}+ u(-v^{-2}v')$$
$$= \frac{u'}{v}+ \frac{uv'}{v^2}= \frac{u'v}{v^2}+ \frac{uv'}{v^2}= \frac{u'v+ uv'}{v^2}$$

 

[haruspex]

$$= \frac{u'}{v}+ \frac{uv'}{v^2}= \frac{u'v}{v^2}+ \frac{uv'}{v^2}= \frac{u'v+ uv'}{v^2}$$

$$= \frac{u'}{v}- \frac{uv'}{v^2}$$ etc.

 

[jgreen520]

Thanks for the response.

So differentiating D-LCos(theta) the constant D goes to 0 and the -LCos(theta) goes to +Lsin(theta) ...then multiply that by sin(theta) in the denominator to get Lsin^2(theta)...

However I was curious why nothing happens to the L...in front of the Cos.

Thanks

 

[HallsofIvy]

"L" is a constant. What do you think should happen to it?